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AOSS 321, Winter 2009 Earth System Dynamics Lecture 4 1/20/2009

This lecture discusses divergence, curl, and the Laplace operator in spherical coordinates, as well as vorticity and its applications in geophysical flows. It also covers the divergence and relative vorticity of wind fields, and the Taylor series expansion.

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AOSS 321, Winter 2009 Earth System Dynamics Lecture 4 1/20/2009

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  1. AOSS 321, Winter 2009Earth System DynamicsLecture 41/20/2009 Christiane Jablonowski Eric Hetland cjablono@umich.eduehetland@umich.edu 734-763-6238 734-615-3177

  2. Today’s lecture We will discuss • Divergence / Curl / Laplace operator • Spherical coordinates • Vorticity / relative vorticity • Divergence of the wind field • Taylor expansion

  3. Divergence(Cartesian coordinates) • Divergence of a vector A: • Resulting scalar quantity, use of partial derivatives

  4. Curl(Cartesian coordinates) • Curl of a vector A: • Resulting vector quantity, use of partial derivatives

  5. Laplace operator(Cartesian coordinates) • The Laplacian is the divergence of a gradient • Resulting scalar quantity, use of second-order partial derivatives

  6. Spherical coordinates • Longitude (E-W) defined from [0, 2]:  • Latitude (N-S), defined from [-/2, /2]:  • Height (vertical, radial outward): r r v 

  7. Spherical velocity components • Spherical velocity components: • r is the distance to the center of the earth, which is related to z by r = a + z • a is the radius of the earth (a=6371 km), z is the distance from the Earth’s surface

  8. Unit vectors in spherical coordinates • Unit vectors in longitudinal, latitudinal and vertical direction: • Unit vectors in the spherical coordinate system depend on the location (, ) • As in the Cartesian system: They are orthogonal and normalized

  9. Velocity vector in spherical coordinates • Use the unit vectors to write down a vector in spherical coordinates, e.g. the velocity vector • In a rotating system (Earth) the position of a point (and therefore the unit vectors) depend on time • The Earth rotates as a solid body with the Earth’s angular speed of rotation  = 7.292  10-5 s-1 • Leads to

  10. Time derivative of the spherical velocity vector • Spherical unit vectors depend on time • This means that a time derivative of the velocity vector in spherical coordinates is more complicated in comparison to the Cartesian system:

  11. Spherical coordinates , , r • Transformations from spherical coordinates to Cartesian coordinates:

  12. Divergence of the wind field • Consider the divergence of the wind vector • The divergence is a scalar quantity • If (positive), we call it divergence • If (negative), we call it convergence • If the flow is nondivergent • Very important in atmospheric dynamics

  13. Divergence of the wind field (2D) • Consider the divergence of the horizontal wind vector • The divergence is a scalar quantity • If (positive), we call it divergence • If (negative), we call it convergence • If the flow is nondivergent Unit 3, frame 17: http://www.atmos.washington.edu/2005Q1/101/CD/MAIN3.swf

  14. Vorticity, relative vorticity • Vorticity of geophysical flows: curl of(wind vector) • relative vorticity: vertical component of defined as • Very important in atmospheric dynamics

  15. Relative vorticity • Vertical component of • Relative vorticity is • (positive) for counterclockwise rotation • (negative) for clockwise rotation • for irrotational flows • Great web page: http://my.Meteoblue.Com/my/

  16. Real weather situations500 hPa rel. vorticity and mean SLP sea level pressure Positive rel.vorticity, counterclock-wise rotation,in NH: lowpressure system

  17. Divergent / convergent flow fields • Compute the divergence of the wind vector • Compute the relative vorticity • Draw the flow field (wind vectors) y x

  18. Rotational flow fields • Compute the rel. vorticity of the wind vector • Compute the divergence • Draw the flow field (wind vectors) y x

  19. Taylor series expansion • It is sometimes convenient to estimate the value of a continuous function f(x) about a point x = x0 with a power series of the form: • In the last approximation, we neglected the higher order terms

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