Coordinate Transformations

Under construction (this web page, and most other web pages for this course).

Instructor:   Roland Stull

Learning Goals:   By the end of this module, you will be able to ...

• explain the motivation for terrain-following vertical coordinates
• explain and plot a hybrid-eta vertical coordinate, as is used in WRF-ARW4
• for total derivatives, gradient of a scalar, and divergence of a vector, be able to convert them to different coordinate systems
• identify the "metric term" in the WRF4 eqs of motion.
• convert lat/lon coordinates on Earth to x/y coordinates on a polar stereographic projection.
• compare different map projections:  pros, cons, and how they are made.
• discuss the relationship between map and earth coordinates, and the role of the map scale factor "m".
• show how terms in the eqs of motion are changed to include map factors.

Readings BEFORE class:

1. read section 3.2.1 (p23-42) in our primary textbook (Warner 2011).
2. re-ead Stull p748, and read Warner  p23-42 and p50-51 on numerical frameworks.
3. IN Class: 
   -Handout showing a regular grid on a polar stereogr. map, and corresponding domain in lat/lon coordinates. 
   -Handout from Haltiner & Williams on map projections. 
   

Homework AFTER class:

1. Read Haltiner & Williams handout.
2. Re-read WRF-ARW Tech note section 2.4 . 
3. Homework #3, due in 1 week or so.
   1) Given this csv file n_hem.csv of lat/lon for N. Hemisphere coastlines (roughly 45,000 data points). Use that file to create two maps, similar to the maps in the solved example on page 748 Stull 2017 Practical Meteorology.  Namely, (a) one map is a plot of the coastlines on a lat/lon grid, and (b) the other is a plot of the coastlines on a polar stereographic grid using 60°N as the reference latitude. For example, here is the (a)  lat/lon plot of that csv file, plotted using RStudio (but please make your own plot, using your own program). See the "caution" statement below.
   2) On page 748 Stull 2017 Practical Meteorology, in the INFO box, is eq. (F20.2), which gives an expression for "r".  This looks different than the expression I gave in class.  Start with the eq for "r" that Stull wrote on the blackboard during class, and show how you can manipulate that equation to get eq (F20.2). 
   3) For the second advection term shown in class (term II), simplify it using equations such as eq (M27) or other equations.  Show your work, and discuss the interpretation of your results.
   
   Caution regarding the coastline file: There are empty rows in the file, to separate the different lines that are plotted. Also, there is a header line. So you might want to view the file in a text viewer or in Excel to see what the file looks like first, to decide how to read it into your own program.  [credits: Thanks to Eve Wicksteed, for finding the original world coastline data at: https://www.evl.uic.edu/pape/data/WDB/ , which is a very large file with millions of data points.]
   
   

Topics

A. Vertical Coordinate Transformations

1. pressure
2. terrain following
3. hybrid

B. Horizontal Coordinate Transformations

1. Map projections & map factors.
   
   • Conformal: Mercator, Lambert, Polar Stereographic.  
2. WRF uses polar stereographic. 
   
   1. Derivation of map amplification factors (m) and coordinate rotations for polar stereographic grids.
   2. Application of horiz. coord. transformations to the equations of motion.  Derivation for some of the terms.
      
   3. WRF-ARW equations on a polar stereogr. grid.