Name _________________________

 

Date __________________________

 

 

 

 

 

 

 

 

 

 

 

Meteorology 430

Fall 2008

 

 

 

Lab 4

 

Absolute Geostrophic Vorticity

 

 

 

 

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

            answer is provided.

 


 

A.  Introduction

 

i.  The Nature of the Problem

 

You have had enough exposure to the theories of surface pressure development to realize that one of the keys to diagnosing the location and intensity of developing surface weather systems is a fairly detailed knowledge of the vorticity patterns in the middle and upper troposphere.  You have learned that in the upper troposphere vorticity advection is of the same order of magnitude as the divergence (local change is small and dropped out on an order of magnitude basis for synoptic scale motions). 

 

Relative vertical vorticity is

 

 

                                                                                                    (1)

 

 

where u and v are the horizontal wind components.  If one had, say, a map of 300 mb winds, such winds can be broken down into there components and the gradients in equation (1) calculated to obtain the relative vorticity of the real wind in a manner analgous to that you used in the previous lab for calculating horizontal divergence.  However, this method also is prone to large errors in the estimates of relative vorticity because the gradients of the horizontal wind components are one or two orders of magnitude larger than the relative vorticity itself.  Thus, the result is very sensitive both to the measurement accuracy of the winds and the estimates of u and v used.

 

ii.  A Way Around the Problem

 

Operational meteorologists often use the 500 mb vorticity and height field to infer the location of vorticity advection patterns in the upper troposphere.  Most forecasters do not even know why such vorticity advection in the middle troposphere (where divergence is nearly zero) can yield useful information about divergence patterns aloft.

 

In Metr 200/201/400, you learned that, above about the 700 mb level, trough and ridge patterns are nearly "vertically stacked" so that the geometry of the 300 mb pattern is nearly identical with the geometry of, say, the 500 mb pattern.  In Metr 403, you have learned why.  Most of the temperature advection in the troposphere takes place beneath 850 mb.  Above the 850 mb level,  ridges and troughs tend to be warm and cold core, respectively, (that is, they are equivalent barotropic) and vertical “stacking” is characteristic.

 

Because of this, the curvature of the height contours and the location of the strongest flow is roughly the same for every constant pressure surface map from around 700 mb to the tropopause.  Since relative vorticity is due both to curvature effects and shear, the locations of vorticity maxima and minima and regions of cyclonic vorticity advection (CVA) and anticyclonic vorticity advection (AVA) are roughly identical for every chart from the 700 mb chart on up.  Thus, the positions of vorticity centers and advection patterns at, say, 300 mb (where divergence/convergence can be anticipated to be very strong) can be inferred from their positions on, say, the 500 mb chart.

 

 

 

Exercise 1:       Contour the 300 mb chart for 12 UTC 20 Oct 1994 given (heights only) at the standard interval.  Compare with the contoured 500 mb chart for 12 UTC 20 Oct 1994.  Verify that the geometry of the pattern (locations of troughs and ridges, jet streams etc.) is similar on both charts.

 

Here are some cautions:

 

1.         Although streamline curvature may be nearly identical (comparing the 500 mb height pattern to that at 300 mb), remember that curvature vorticity is the product of the wind speed and the curvature.  Since wind speeds are stronger in the upper troposphere, the magnitude of the curvature vorticity will be greater.  Also, since wind speeds at the core of the current are so much stronger in the upper troposphere, shear vorticity will also have greater magnitude in the upper troposphere. Thus, only the shape of the vorticity patterns will be nearly identical.  The strength of the maxima and minima and, also, the vorticity gradients will be greater in the upper troposphere.

 

2.         Remember that horizontal divergence is proportional to the product of the wind speed and the absolute vorticity gradient divided by the absolute vorticity at the grid point at which an estimate of the divergence is to be made.  Since wind speeds in the upper troposphere tend to be greater than those at 500mb, vorticity advection has a greater magnitude in the upper troposphere.

 

The result of these cautions is that you should only use the CVA/AVA patterns at 500 mb to diagnose WHERE  upper divergence and convergence can be expected to occur for a given pattern and only the RELATIVE STRENGTHS of the divegence and convergence.  The question remains, why are we using the vorticity patterns at 500 mb and not 700 mb (or any other level, for that matter)?

 

The relative and absolute vorticity of the geostrophic wind can be obtained directly from the height field without the analyst going through the intermediate step of specifying the gradients of the wind components.  (You will derive the appropriate equation below).  Since the level at which  the real wind is most nearly geostrophic (non-divergent) is the 500 mb level, the absolute geostrophic vorticity field is an accurate representation of the real vorticity field at that level.  Thus, the vorticity advection patterns at 300 mb (or, at any level, for that matter) can be qualitatively diagnosed by the patterns of geostrophic vorticity advection at the level of nondivergence (which, we assume, is near the 500 mb level).

 

In this exercise, you will learn how closely the 500 mb vorticity patterns which appear on the NGM initialization corresponds to patterns of absolute geostrophic vorticity which can be obtained simply from the 500 mb height pattern.  Also, you will derive the finite difference version of the relative geostrophic vorticity and use this form of the expression as the basis of a short computer program to run on a personal computer.

 

 

Question 1:      Substitute the geostrophic wind equations (in x,y,p) coordinates into equation (1) and expand.  Assume that the Coriolis parameter is constant.

Geostrophic wind equations

 

                                    Simplify using

 

 

                                                                                  (2)

                                   

 

            where (2) is the horizontal Laplacian and provides a quantitative estimate of the shape of the field in question (in this case, the heights).  The Laplacian of the height   field is an estimate of the variation of the slope of the height field along the horizontal coordinate axes.

 

 

 

Note the finite difference grid below.  The crosses below indicate grid points at which heights are recorded.  The grid points are labeled (unconventionally) 0,1,2,3,4,A,B,C and D and are all located at distance of "s" from the adjacent grid point.

 

 

 

 

The derivative  can be evaluated at point A by the finite difference expression

 (Z1 - Z0)/d and at point C  can be approximated by the expression (Z0 - Z3)/d.

 

 

Metr 430 -- Lab #4 -- Page 4

 

The derivative  can be obtained by subtracting the height gradient at C from that at A (both obtained above) and dividing by the distance between A and C.  The result is the finite difference approximation for the term furthest to the right of the equals sign in the equation you developed in Question 1 above.

 

 

Question  2.     Perform the same derivation for the first term to the right of the equals sign in the equation you developed in Question 1 above.

 

                        Algebraically add the results in this section to obtain the full finite difference

            equivalent for the equation you developed in Question 1 above.

 

 

The finite difference equation you developed above states that the relative vorticity is directly proportional to the shape of the height field as estimated for the variation in slope of the height surface along the two coordinate axes.

 

 

                                                 (3)

 

You are nearly ready to compute absolute geostrophic vorticity from the map of 500 mb data attached.  However, to compute absolute vorticity one needs to know the value of the Coriolis parameter at the same range of latitudes as given above.

 

 

Question 3. Compute the value of "f" at each of the latitudes given  below.

 

            Latitudes                                Value of "f"

            27.5

30

            32.5

            35

            37.5

            40

            42.5

            45

            47.5

 

 

The equation that you developed states that the 500 mb absolute geostrophic vorticity (hereafter called absolute vorticity) can be obtained if the analyst can obtain 500 mb heights at each of the five grid points.  Numerical schemes can do this directly from the upper air data gridded on the basis of information from the radiosonde sites.  The 500 mb heights are interpolated to the grid points using various objective schemes.

 

 

  The analyst can perform an analagous procedure if he or she is presented with a map of uncontoured 500 mb heights in the field.  A careful contouring of the data at 3 decameter intervals can lead to adequate estimates for the 500 mb heights at the grid points.  The contours are meticulously constructed making sure they are oriented correctly with respect to the wind field (wind flow parallel to the contours and contour spacing inversely proportional to the wind strength).

 

 

Exercise 2.       For the 500 mb chart given,  complete the vorticity     spreadsheets provided.  Use exactly the same procedure as in the previous lab with the same transparancy overlays to compute the vorticity.

 

Work in teams, as discussed in class.

 

First, place a blank acetate on the 500 mb chart.   Draw a cross centered at 47.5N, 95W that has dimensions of d=2.5o latitude (careful, this will not correspond to 2.5 o longitude due to convergence of the meridians).

 

Second, calculate the dimensions of d in meters.

 

Third, label points 1, 2, 3, 4 and 0 on your cross. 

 

Fourth, now place the grid on top of the analyzed 500 mb chart.  On top of that, overlay the acetate with the finite difference cross at the intersection of 45N with 95W.  Obtain the value of

 

 

at each latitude longitude intersection across the range of latitude from 27.5N to 47.5N and longitudes 95W to 75W, as explained in class.

 

Be sure to convert to relative vorticity and then to absolute vorticity by using g and the value of f at that latitude (from the table above and recorded in the appropriate location on the spreadsheet).  Remember to convert the heights from decameters to meters.  Record right on the 500 mb chart under the center point of the grid.

 

 

 

 

 

 

 

 

 

Exercise 3.  Once the values are obtained, each student will contour the absolute vorticity field at 2 X 10-5 s -1 intervals. Compare and contrast your chart with the NGM initial analysis of the same 500 mb height field and wxp’s analysis of the vorticity field at 300 mb (keeping in mind the background information on pp. 1 and 2 above).