Name _________________________

Date __________________________

Meteorology 403

Fall 2008

Lab 3

Divergence

1.      All labs are to be kept in a three hole binder.  Turn in the binder when you have finished the Lab.

2.      Show all work in mathematical problems.  No credit given if only

A.  General

In this lab you will compute horizontal divergence at the 300 mb level by finite difference approximation of the derivatives ∂u/∂x and ∂v/∂y.

B.  Case Study

You are provided with the following maps for 00 UTC 28 September 1994:

1.  300 mb analysis with isotherms and isotachs.

2.  Unanalyzed 300 mb chart with raw data.

3.  300 mb “u wind component”

4.  300 mb “v wind component”

5.  NGM Initialization of 500 mb heights, Absolute Vorticity

6.  NGM Initialization of Surface Pressure, 1000-500 mb thickness.

7.  300 mb wind vectors with data domain and analysis grid for this exercise.

C.  Nature of Analysis

As Bluestein points out, horizontal divergence is the sum of two small derivatives that often have opposite sign.  Finite difference approximation of the derivatives often shows that the terms themselves may be one or two orders of magnitude larger than the net divergence when both are added algebraically.  Thus, wind speed components would need to be accurate to two or three places in order for the resulting divergence estimates to be accurate.

It is still a useful exercise to compute divergence from the expression in rectangular coordinates and compare with the actual divergence field as produced by the wxp programs.  The case chosen is for a situation in which a strong jet streak was moving around a relatively strong trough over the eastern United States.

D.  General Procedure

We are going to use a relatively coarse analysis grid.  Each point at which we would like to have divergence estimates is 2.5 degrees of latitude and 5.0 degrees of longitude from the adjacent point.  It will be interesting to see if your analysis captures the “synoptic scale flavor” of the wxp field.

We also will be using a so-called “centered difference” approach in evaluating the derivatives.  That simply means that the analysis point is at the origin of the finite difference cross and the end points are a certain distance +/-∆s along the x and y axes from the origin.

The finite difference approximations of the derivative ∂u/∂x are obtained at each of the analysis grid points and the values are then plotted in black near the analysis point.  The finite difference approximations of the derivative ∂v/∂y are then obtained at each of the same analysis grid points then are plotted in blue at each respective point.  The algebraic sum of the two should be plotted in green and then contoured at intervals of 2 X 10 -5s-1 and then transferred to the 300 mb wind vector chart (with the analysis domain).

To accomplish the calculations the interval ∆s must be selected.  To do this, lay a blank acetate on the analysis point at 47.5N, 95W.  Draw a finite difference cross with the origin at the point 47.5N, 95W.  The the interval ∆x should correspond to 10 degrees of longitude AT 47.5N.  The interval ∆y will always correpond to 5 degrees of latitude.

Distance in KM of 10 degrees of longitude = _____________  = ∆x

Distance in KM of 5 degrees of latitude = _____________  = ∆y

In black, draw your cross on the acetate, as shown in class.  You will then simply move your cross from analysis point to analysis point and compute the derivatives on the basis of the values at the ends of the axes, as discussed in class.

Students will work cooperatively to obtain ∂u/∂x and ∂v/∂y:

E.  Thought and Other Questions

1.      Compare and contrast your divergence analysis with that of wxp (distributed to class separately from the lab).

2.      Contour the unanalyzed 300 mb charts for heights and isotachs using conventional intervals.  Do NOT ask.  Covered in Metr 201/400.

3.      (a)     Estimate the qualitative nature of the SYNOPTIC SCALE divergence on the basis of the vorticity advection shown on the                      NAM initialization for the same area as your contour chart.

(b)     Estimate the qualitative nature of the forcing for vertical motion on the basis of temperature advection shown on the NGM                     initialization for the same area.

 Please remember that part of the learning experience is learning to deal with the perils, pitfalls and strengths of cooperation.  The idea here is to have the students themselves deal with the micro-management of the task, just as you will have to when assigned group tasks in the real world.  The instructor (or overseer, or boss) should not have to deal with petty squabbles, and should only be asked to adjudicate significant issues.

Each group should select an overseer, who should also participate in the calculations and other tasks.  Each group should select a spokesperson who will interact with me if problems come up.  Each group should work in tandem to perform the calculations.