Homework: Bluestein Quasi-geostrophic Brain Teasers
The key both to answering the question posed correctly and to providing an
organized, coherent summary of the answer to the class was a structured approach.
In essence, the structured approach to problem solving now in your senior year (or
at graduate level) is the same as outlined by Dave Dempsey in Metr 201. It would
be modified to answer the questions posed because numerical calculations and
answers were not required. The best way to approach this is to start with a general
principle and then work your way to the specifics.
Once the general principles that govern the way to a solution are obtained, each
can be simplified on the basis of the simplifications that Bluestein mentions.
Each needs to be implicitly or explicitly explained, and not dismissed with
“Bluestein says to neglect…” The grading was based upon how carefully the
presenter explained this or made it obvious, without just stating it as a fact that
does not need to be explained.
The grading also was made taking into account the fact that the presenters were, in
effect, teachers. I also am definitely assessing your abilty to “teach” the class what
you did. If you were stumbling around, you got points off. Incorrect use of
terminology at the senior level is not allowed and the point totals reflect that.
I’ve assigned difficulty factors based upon increasing difficulty of the problem.
The most difficult problem was (a) the Quasigeostrophic Query, the second most
difficult,(b) the Vorticity Vexation and the the third most difficult (c) the
Convergence Conundrum. Extra points assigned because of the difficulty issues
were (a) 1; (b) 3; (c) 2.
Convergence Conundrum
A. Organizing Principles and Relations
1. Dine’s Compensation
The organizing principle relating vertical motion in the mid-troposphere to surface
convergence or divergence was Dine’s Compensation. With the boundary
conditions of zero vertical velocity at the ground, surface convergence could be
inferred where 500 mb vertical velocities were upward. (2 points) (Emily 0; YiLin
2, Geoff 2)
2. QG Omega Equation
500 mb vertical velocities could be deduced by application of the quasigeostrophic
omega equation’s four forcing terms. By the constraints listed in the problem, the
last two terms drop out (diabatic heating and latent heat release related to
condensation of water vapor are embedded in Term C and frictional effects in
Term D). Thus, YiLin and Emily needed to carefully consider the differential
vorticity advection term and the temperature advection term. (2 points) ) (Emily 2;
YiLin 2, Geoff 2)
Considering the temperature advection forcing term, one must carefully lay out
that if the geostrophic wind is zero at the surface the thermal wind is the 500 mb
geostrophic wind and, hence, the isotherms/thickness contours would be parallel to
the height contours, so there would be no contribution from the temperature
advection term. A key part of this was the student’s ability to verbalize the exact
nature of the pattern and to communicate that to the audience (3 points) (Emily 3;
YiLin 3, Geoff 3)
The differential vorticity advection term can be assessed from the 500 mb vorticity
advection because in the constraints for the problem there is zero wind speed at the
ground, and the sign of the 500 mb vorticity advection can be used to infer the sign
of that term. (3 points) ) (Emily 3; YiLin 3, Geoff 3)
B. Qualitative Calculation
The next step is to correctly assess the vorticity field in order to infer the areas of
CVA and AVA. Embedded in this step is that one needs to recognize that the
pattern depicted is an anticyclone. Since the pattern was the clearest of the three
problems (meaning actual heights with no other complicating factors) I did not
include the figure in the discussion below (1 point) ) (Emily 1; YiLin 1, Geoff 1)
Once the fact that the pattern shown in Fig. 1.24 is an anticyclone, one must infer
the absolute vorticity field associated with it. Absolute vorticity is the relative
vorticity plus the coriolis parameter, but the pattern is across a small range of
latitudes so the absolute vorticity advection really relates directly to relative
vorticity advection. (3 points) ) (Emily 2.5; YiLin 2.5, Geoff 3)
Relative vorticity is due to shear and to curvature. Since the 8 locations shown are
in the core of the jet, the shear term drops out. (2 points) ) (Emily 2; YiLin 2,
Geoff 2)
The curvature at every location is anticyclonic. But the curvature is “most”
anticyclonic at locations 1 and 5, and “least” anticyclonic at 3 and 7. Thus, there is
anticyclonic vorticity advection at locations 4 and 8 and cyclonic vorticity
advection at locations 2 and 6. (2 points) ) (Emily 2; YiLin 2, Geoff 2)
Therefore, the quasigeostrophic omega equation “diagnoses” upward vertical
velocity at locations 2 and 6. Dine’s Compensation then can be used to infer
surface convergence at the same locations. (2 points) ) (Emily 2; YiLin 2, Geoff 2)
Grades: Emily 17.5/20; YiLin 19.5/20; Geoff 20/20 +1 Difficulty= Emily
18.5/20; YiLin 20.5/20 and Geoff 21/20.
The Quasigeostrophic Query
A. Organizing Principles and Relations
1. Dine’s Compensation
The general organizing principle relating vertical motion in the mid-troposphere to
surface pressure tendency was Dine’s Compensation in the form of the pressure
tendency equation. With the boundary conditions of zero vertical velocity at the
ground, surface divergence and pressure rises could be inferred where the vertical
velocities at the top of the layer bounded by the ground were downward.
However, the complicating factor was that an imposed vertical motion field due to
topography was also present. The effect of that, quasigeostrophically, is do induce
surface convergence were topotraphically generated subsidence is occurring and
surface divergence were topographically generated upward motion is occurring.
This was the most difficult problem to conceptualize because of the number of
different factors at play. In addition, the pattern depicted was the most bizarre of
the three patterns assigned. (4 pts) (Sierra 2, Ryan 2, Malori 2, Geoff 4)
2. QG Omega Equation
The vertical velocities could be deduced by application of the quasigeostrophic
omega equation’s four forcing terms. By the constraints listed in the problem, the
last two terms drop out (diabatic heating are embedded in Term C and frictional
effects in Term D). Thus, Denise and Lexee needed to carefully consider the
differential vorticity advection term and the temperature advection term and then
integrate in the topographic effects. (2 points) (Sierra 2, Ryan 2, Malori 2, Geoff 2)
B. Qualitative Calculation
The temperature advection is easy to determine, given the bizarre configuration of
the temperature contours. There is warm advection on the east and warm
advection on the west so this term alone would contribute to forcing for pressure
rises in the west and falls in the east. (2 point) (Sierra 2, Ryan 2, Malori 2, Geoff
2)
The vorticity advection term is much more difficult to conceptualize and involves
adding the thermal wind vector to the 1000 mb geostrophic wind vector
qualitatively. The thermal wind would be blowing parallel to the bizarre
configuration of the temperature contours given (warmer air to the north). This
leads to the suggestion of a ridge axis right over the center of the surface pattern,
with CVA to the east and AVA to the west. This would lead to pressure rises in
the west and falls in the east. The grading here had to do mostly with recognizing
the strangeness of the pattern and applying the thermal wind correctly (4 points)
(Sierra 4, Ryan 4, Malori 4, Geoff 4)
So far, the determination would be that the anticyclone would move or develop
towards the west. However, the last thing to be considered is the effect of the
topographically generated vertical motion, which has the effect of increasing
pressures on the upslope side and decreasing pressures on the downslope side of
the flow. The way I graded here centers on if you made at least an attempt at
applying the principle, even though we didn’t discuss it in class (3 points) (Sierra
2, Ryan 2, Malori 2, Geoff 3)
The net forcing could be obtained by conceptually adding all three effects. That’s
what I did to get this. My results suggest that the high pressure area should drift
northwestward, independent of the magnitudes of the different terms, because all
three terms are in phase in two locations. (5 points) (Sierra 4, Ryan 4, Malori 4,
Geoff 5)
Grades: Sierra 16/20; Ryan 16/20; Malori 16/20; Geoff 20/20 +3 Difficulty
= Sierra 19/20; Ryan 19/20; Malori 19/20; Geoff 23/20
The Vorticity Vexation
A. Organizing Principles and Relations
1. QG-Height Tendency Equation
The problem asks the reader to find regions on Fig. 1.26 that would experience
positive height tendencies. Unlike the other two problems, this one can be
approached directly using the quasigeostrophic height tendency equation. There
are four terms, two of which, those associated with temperature advection, diabatic
effects and friction drop out, because of the constraints of the problem. The reader
must understand why the temperature advection term drops out because the
atmosphere is barotropic. Some attempt must be made to explain that.
(4 points) (Andrew 2; Craig 2; Geoff 4)
2. Visualization
While the equation can be qualitatively assessed directly from the QG-height
tendency equation, this question tests the ability of readers to visualize the actual
nature of the fields and rewards those who do NOT use mere pattern recognition.
At first glance at Fig. 1.26, readers using pattern recognition will see a
conventional ridge and trough in the westerlies. Only those who look closely at the
heights would note that that is in fact true for the northern half of the chart, but on
the southern half, heights decrease southward.
(4 points) (Andrew 4; Craig 4; Geoff 4)
3. The Vorticity Advection Term
This is the centerpoint of the presentation. The letters on the diagram are in the
region of greatest flow, at the core of the jet. Thus, the student is justified in
leaving out shear vorticity. Otherwise, you complicate the problem, though it can
still be done.
(a) For wavelengths <<<6000 km
For very long wavelengths (at a given amplitude) the curvature of the contours
relaxes so much that the cyclonic vorticity advection downstream from curvature
maxima or minima decreases to negligible levels. For very short wavelengths, the
CVA/AVA related to the relative vorticity field is very, very large. Thus, the
advection of planetary vorticity can be neglected on an order of magnitude basis.
Key in getting this part correct is recognizing the weirdness of the pattern (on its
southside). The areas of CVA are downstream from maxima, which are in a
different location on the “upside down” southern portion of the pattern. The same
is true for the areas of AVA.
The QG-height tendency equation thus diagnoses height rises at locations J, M, O
and L.
(b) For wavelengths >>>6000 km (We hadn’t gone over this, so I didn’t grade
for this)
For very long wavelengths (at a given amplitude) the curvature of the contours
relaxes so much that the cyclonic vorticity advection downstream from curvature
maxima or minima decreases to negligible levels. For very long wavelengths, the
CVA/AVA related to the advection of planetary vorticity field dominates the
advection of relative vorticity. Thus, one can neglect the advection of relative
vorticity advection on an order of magnitude basis.
Key in getting this part correct is recognizing the weirdness of the pattern (on its
southside). The areas of CVA are downstream from maxima, which are in a
different location on the “upside down” southern portion of the pattern. The same
is true for the areas of AVA. The problem also is that the earth vorticity increases
with latitude.
The QG-height tendency equation thus diagnoses height rises at locations G, P, L
and O.
In this grading section, I am quantifying the way terminology was used (calling
heights, heights and not winds etc.) and proper assessment of terms (knowing
where cyclonic vorticity advection was and what it means). I also am definitely
assessing your abilty to “teach” the class what you did, as pointed out above. I did
not grade off for systematic error (because you didn’t recognize the upside down
portion of the pattern).
(12 points) (Andrew 8; Craig 10; Geoff 12)
Grades: Andrew 14/20, Craig 16/20; Geoff 20/20) +2 Difficulty= Andrew
16/20, Craig 18/20 and Geoff 22/20.