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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

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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

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(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

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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.

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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

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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).

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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

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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

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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

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Figure 9

Figure 10

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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

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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

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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

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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.