1
An Analysis of 500 hPa Height Fields and Zonal Wind: Examination of the Rossby
Wave Theory
Justin Hayward, Chris MacIntosh, Katherine Meinig
Department of Geologic and Atmospheric Sciences, Iowa State University, Ames, IA
ABSTRACT
This project studied the 500 hPa height fields and zonal wind in order to test if the observed data correlates
with the Rossby Wave Theory. Data collected includes wave number, speed, amplitude and zonal wind
speed through 1 September – 11 November 2011.Graphs are included to show evolutions over the time
period and correlations between different components. Results showed a slight correlation to the Rossby
Wave Theory, but with little significance in most cases. Two methods to reduce the errors and assumptions
would be to use a longer data set or to find an atmosphere that is completely barotropic.
1. Introduction
Large-scale waves are important not only to meteorologists, but to people worldwide.
Their patterns allow for variations in weather across regions. By understanding these patterns
and their components such as 500 hPa wind speeds and wave amplitude, meteorologists are able
to better forecast future events. With the ideas behind the Rossby Wave Theory, the relationship
can be found between different elements of waves. These elements can range anywhere from the
way that amplitude affects wave speed to differences between wave patterns in the Northern
Hemisphere and the Southern Hemisphere. The combination of how these components work
together gives meteorologists a better idea of how to forecast synoptic patterns more accurately.
This paper will look at the methodology of the research, the results achieved and the overall
analysis of the waves in the Northern and Southern Hemispheres.
2. Data and Methodology
The data used in this research was taken from the Iowa State University Weather
Products website (http://www.meteor.iastate.edu/wx/data). This data looks at the 500 hPa level
in the Northern and Southern Hemisphere as well as the zonally averaged winds. The data used
to produce this output was taken at 00 UTC each day, and during this project we analyzed this
data over a 72-day period. The data analyzed were wave number (N), wave amplitude (A), wave
speed (C), 150-300 hPa maximum zonal wind, and 500 hPa average zonal wind in both the
2
Southern and Northern Hemispheres. When calculating the wave number, amplitude, and speed
we looked at plots of 500 hPa heights using the 50 degree north and south latitude circle. All of
our data was inputted into a Google Doc spreadsheet, so that the information could be easily
shared between each group member.
The following guidelines were given to each group in order to standardize the procedure
to attain results throughout the class. As a standard method to measure wave number, we used a
common contour at 5580 m in the Northern Hemisphere and the 5280 m contour in the Southern
Hemisphere. Whenever these contours passed over the 50 degree latitude circle in both
hemispheres, we counted the amount of times it crossed over and divided by two to get the
amount of waves in each hemisphere on a given day. For example, when we counted a total of 6
crossings over the latitude circle, our N=3. There were certain times when the contours would
pass just over the 50 degree line and were included in our collection of wave numbers.
Amplitude was the next set of data we were to attain from these 500 hPa maps. Per a
given set of waves, there were indicated wave maximums and wave minimums. There was a
local maximum and minimum per each wave number. Thus, amplitude could be calculated from
the following:
(1) A = (Avgmaxima / Avgminina) / 2,
where Avgmaxima is the average of the N local height maxima and Avgminima is the average of the
N local height minima.
We also were able to find wave speed (C) from the given 500 hPa maps. Wave speed was
determined by recording the longitude of the local maximum where it crossed over our latitude
circle from the preceding and following days. Using this data, we could find the average wave
speed given a certain day provided this equation:
(2) C = {LON(day+1) - LON(day-1)} / 2,
where LON(day+1) is the recorded longitude of the day after and LON(day-1) is the recorded
longitude of the day before. Thus, wave speed is one half of the change in longitude between the
day after and the day before.
Lastly, given plots of zonal wind per pressure, we were told to analyze and record the
average zonal wind at 500 hPa (U500) and the maximum zonal wind between 150 and 300 hPa
3
(Uupper). Basing our observation using our previous 50 degree latitudes, we recorded the wind
at each of those pressures for both the Northern and Southern Hemisphere.
3. Analysis
a. Average wave speed in Northern and Southern Hemispheres
Over the course of our 72-day timeframe, wave speeds for both the Northern and
Southern Hemispheres were calculated for each day and time period averages for both
hemispheres were found. For a single day maximum, the Northern Hemisphere saw a value of 11
deg day-1
in a west to east motion, and the Southern Hemisphere saw a value of 15 deg day-1
in
an east to west motion. In terms of averages, we discovered that the Southern Hemisphere had a
higher average wave speed (7.98 deg day-1
) than the Northern Hemisphere (5.91 deg day-1
). Both
the single day maximums and averages seem to make sense, since there is less land mass in the
Southern Hemisphere to affect the speed of the waves, i.e. limited topographical effects that
allow the waves to move at a faster rate. At these average speeds, a wave would take 60.91 days
to move around the 50 degree latitude circle in the Northern Hemisphere and 45.11 days to do
the same in the Southern Hemisphere. As expected, the waves moved counter-clockwise in the
Northern Hemisphere and clockwise in the Southern Hemisphere, due to the pronounced effect
of the Coriolis force.
b. Wave speed vs. mid- and upper-level zonal winds
Comparing wave speed with the mid- and upper-level zonal winds can show us a
relationship between the two. However, zonal winds are shown in units of meters per second,
whereas wave speed is in units of degrees per day. In order to do this comparison, the zonal
winds had to be converted to degrees per day.
Figure 1, on the next page, shows the wave speed, plotted on the y-axis, compared to the
500 hPa average zonal wind, plotted on the x-axis, for the Northern Hemisphere. From the graph,
it can be seen that there is a very slight increase in wave speed with 500 hPa average zonal wind;
however, as shown by an R2 value of only 0.003, this result is not statistically significant. This
4
Figure 1
positive correlation does relate to the Rossby Wave Theory, since wave speed is expected to
increase with 500 hPa zonal wind speed. This can be shown by the phase speed equation:
(3) c = ū – β / (k2+l
2),
where ū is the outside wind speed, β = ∂f/∂y is the change in absolute vorticity in the meridional
direction, and k and l are horizontal wave numbers that are dependent on the wave numbers in
the meridional and zonal directions, respectively.
An average of the 500 hPa zonal wind in the Northern Hemisphere was computed as well
and found to be 8.28 deg day-1
. This value is higher than the average wave speed found above
(5.91 deg day-1
), which corresponds to the Rossby Wave Theory, since the air in the waves tends
to move faster than the actual wave, e.g. jet streaks in the overall jet stream.
Figure 2, on the next page, shows the wave speed, plotted on the y-axis, compared to the
500 hPa average zonal wind, plotted on the x-axis, for the Southern Hemisphere. From the graph,
there, once again, seems to be an increase in wave speed with 500 hPa average zonal wind speed.
As stated above, this relates to the Rossby Wave Theory; however, this finding also has a small
R2 value (0.027) and is not statistically significant.
5
Figure 2
The average 500 hPa zonal wind was calculated for the Southern Hemisphere as with the
Northern Hemisphere, and was determined to be 10.35 deg day-1
. This value is also higher than
the average wave speed found above (7.98 deg day-1
), which, once again, corresponds to the
Rossby Wave Theory for the same reason stated above for the Northern Hemisphere.
In Figures 3 and 4 on the next page, the wave speed was plotted against the 150-300 hPa
maximum zonal wind (converted to deg day-1) to determine if there was any sort of dependence
of wave speed on upper-level winds. For both the Northern and Southern Hemispheres, there is a
weak positive correlation, which would imply that as the upper-level winds increase, wave speed
also tends to increase. However, the results are clearly not significant and thus, no conclusion
can truly be drawn from this. This lack of dependence can be explained by the interaction
between the upper-levels and the mid-levels, with the upper-level wind speed having little to no
effect on the mid-level wind speed or, in this case, wave speeds.
c. Wave speed without zonal wind vs. wave number
6
Figure 4
Figure 3
In order to properly compare wave speed to wave number, the zonal wind at 500 hPa
must be subtracted to remove all outside influence from the wave speed (recall the phase speed
equation in 3b). This calculation was done for both the Northern and Southern Hemisphere with
both the 500 hPa zonal wind and wave speed being used in units of degrees per day. Graphs of
the wave speed without the zonal wind against the wave number were created for each
hemisphere and are shown in Figures 5 and 6 (next page).
7
In the Northern Hemisphere (Figure 5, below), the wave speed is decreasing with an
increase in wave number, agreeing with the Rossby Wave Theory. This is due to the wave
number being in the denominator of the phase speed equation, i.e. as wave number increases,
phase speed will decrease. Although this finding is consistent with this wave theory, the
correlation is very weak (R2 of 0.002) and cannot be ruled significant.
In the Southern Hemisphere (Figure 6, also below), the wave speed is actually increasing
with wave number, which does not agree with the Rossby Wave Theory. This result is more
significant than in the Northern Hemisphere; however, the R2 value is still not significant
(0.055).
Figure 5
Figure 6
8
d. Wave number over the 72-day period
The Northern Hemisphere (Figure 7, below) has seen a gradual decrease of the number of
waves since the beginning of the time period. With a R2
value of 0.017, there is not a significant
correlation with the wave number as the fall season progresses. Yet, there seems to be a
consistent upward and downward trend that spans over an average ten day period. There were
also a few periods where a consistent number of waves was seen over an extent of a few four day
time frames. Comparatively, these wave patterns are less than the typical synoptic time scale
(around ten days).
Alternatively, the Southern Hemisphere (Figure 8, next page) has seen a greater decrease
in wave number over time. A R2
value of 0.28 suggests a much stronger correlation than that of
the Northern Hemisphere. As with the Northern Hemisphere, the Southern Hemisphere saw a
general upward and downward trend spanning an average six day period. This shorter time frame
could be attributed to reduced land mass in the Southern Hemisphere. There are two time periods
in early September and late October where the wave number remains the same for eight and
seven days respectively. These two wave pattern durations are similar to the length of the
synoptic time scale.
Figure 7
9
e. Amplitude vs. wave number
Based on our previous knowledge of short and long waves, we can expect that shorter
waves will be associated with longer amplitudes and longer waves will be associated with shorter
amplitudes. In the Northern Hemisphere (Figure 9, next page), the data greatly agrees with this
knowledge. A R2
value of 0.28 implies this strong correlation between the amplitude and wave
number. As the time period advances into November, there is a steady increase of the number of
waves with a respective decrease of amplitude. We believe this is true since it is not plausible for
the atmosphere to support multiple deep troughs or high ridges within the same time frame.
The Southern Hemisphere (Figure 10, next page), although not as strongly, agrees with
this relationship. The data offers an R2
value of 0.029 indicating a weak correlation between
amplitude and wave number. As the time period progresses, the amplitude slightly decreases as
the number of waves somewhat increases. The sharp difference in topography, along with a
weaker temperature gradient, in the Southern Hemisphere could account for the difference in
variance between the two hemispheres.
Figure 8
10
Figure 9
Figure 10
11
f. Amplitude vs. date
Based on the plotted graph below (Figure 11, below), there appears to be a gradual
increase overall in wave amplitude. This increase in amplitude comes out to being around 30
degrees after the selected time period is over. Though the increase is visually present in this
graph, the R2 variable (0.09) claims that this correlation is relatively insignificant. We would
expect that wave amplitude would generally increase in the winter months as more significant
troughs are generated and move through the Northern Hemisphere. On average, there are slight
episodes of growth and dissipation in the Northern Hemisphere of amplitude over a period of
five days towards the beginning. These episodes lower to three days or less towards the end of
our time period.
As we look at the plot of amplitude vs. date in the Southern Hemisphere (Figure 12, next
page), we still see a general increase in overall amplitude as time progresses. The increase is a bit
more subtle in the Southern Hemisphere, coming out to about an estimated 10 degrees over the
selected time period. Again, the R2 variable of 0.02 still states that this correlation is rather
insignificant. The presence of this land mass and varied topography can have a profound effect
Figure 11
12
on the motion and large variance in wave amplitude. Inversely, the episodes of growth and
dissipation in the Southern Hemisphere of amplitude occur over a period of three days to start.
These episodes increase to five days or more as the Southern Hemisphere moves into their
Summer Solstice. As stated before, the reasoning behind why the correlation is far less in the
Southern Hemisphere is due to the increased presence of land mass in the Northern Hemisphere.
g. Evolution of zonal winds at 500 hPa and amplitude over time
Toward the beginning of the time period in the Northern Hemisphere (Figure 13, next
page), there is around an eleven day run where the 500 hPa zonal wind averages to 2.5 deg day-1
.
As time progresses, winds experience a general upward and downward trend. The average 500
hPa zonal wind speed for the 72-day day series is 8.28 deg day-1
. The Northern Hemisphere
experiences an overall increase of 500 hPa zonal wind over the time period. It has a R2
value of
0.0075 that supports this slight trend. The Southern Hemisphere (Figure 14, next page) sees a
gradually increasing upward and downward trend throughout the time frame. Unlike the
Northern Hemisphere, the Southern Hemisphere experiences an eleven day stretch at the end of
Figure 12
13
the period where winds remain constant at 0 deg day-1
. With this stretch, the R2
value comes out
to be practically zero showing no correlation with wind speed and time. 500 hPa zonal winds in
the Southern Hemisphere average out to be 10.35 deg day-1
for the whole time period.
Figure 13
Figure 14
14
As stated before in 3f, amplitude in the Northern and Southern Hemispheres see a gradual
increase throughout the time period. R2
values of 0.102 and 0.022 in the Northern and Southern
Hemisphere respectively support this increase. Both hemispheres show a very minor correlation
between the 500 hPa zonal wind speed and amplitude. There are phases where the 500 hPa zonal
wind speed increases as the amplitude increases. This relationship is more pronounced in the
Southern Hemisphere than in the Northern Hemisphere.
4. Conclusion
After analysis of the data that was collected over the course of this fall semester, we have
determined that the Rossby Wave Theory does not seem to be accurately shown in the real
atmosphere. Although most parts of the data, agree with theory, the results are not statistically
significant due to small R2 values; exceptions, however, did occur with wave number vs.
amplitude in the Northern Hemisphere and the time series of wave number in the Southern
Hemisphere with both of them having R2 values of 0.28. This lack of significance can be
attributed to a combination of human error in data collection and assumptions made in the
theory. Rossby Wave Theory assumes a barotropic atmosphere, which is typically not plausible
in the real world, since temperature plays a large role in shaping the world’s weather. However,
the lesser amount of land mass in the Southern Hemisphere allows for the atmosphere to be more
barotropic and thus, the correlations here are slightly greater than in the Northern Hemisphere.
Even though these relationships are slightly more significant in the Southern Hemisphere, it still
cannot be stated with confidence that the Rossby Wave Theory accurately represents the real
atmosphere. If data collection was done over a longer period of time or during a time where the
atmosphere was fairly barotropic, the results may provide a better relationship to the theory and
may show significance.