Atomspheric Waves at the 500hPa Level

Atomspheric Waves at the 5hPa Level Justin Deal, Eswar Iyer, and Bryce Link ABSTRACT Our study observes and examines large scale motions of the atmosphere. More specifically it examines wave motions at the 5 hpa level, including wave number, wave amplitude, and wave speed. It also examines zonal wind patterns in both the northern and southern hemispheres at both 5 mb and between the 3 hpa and 15 hpa layer. Data was recorded between August 29 th, 212 and November 7 th, 212 and placed on time series graphs so comparisons could be made between the two hemispheres and seasonal trends could be observed. I. Introduction Synoptic scale waves are a fundamental mechanic of our planet s atmosphere, and understanding their structure and function is critical. Atmospheric waves are a trigger for many types of weather, so being able to predict their evolution over time is important for not only short term forecasts, but seasonal variations in a regions weather patterns as well. In both the northern and southern hemispheres, two primary features can be observed at any given time. Feature one, a ridge, is usually associated with warm temperatures and mild weather at the surface. Feature two, a trough, can be associated with cooler weathers and precipitation at the surface. Forecasting the evolution of these features alone gives an idea of what one could expect on any given day. Several factors affect the evolution and structure of troughs and ridges. Wave amplitude can be associated with the strength of a system. If a trough is deeper, stronger vertical motions can be expected in front of it, which would lead to enhanced threat of precipitation there. Likewise, with ridges, higher amplitudes will lead to warmer temperatures which reach more towards the poles. The wave number is simply the amount of discernible waves stretching around the globe. Typically ranging from two to six, the wave number can be used to forecast how often weather conditions will be changing. If the wave number is larger, more troughs and ridges will be traversing the globe, which would lead to more frequent changes in weather conditions at the surface. This can be combined with the wave speed (simply the speed at which waves travel across the globe) to develop a fairly accurate forecast. In addition to waves, global zonal winds can be used to gather insight on the global wave pattern. Higher speeds tend to be associated with numerous power systems, which result from high wave numbers of deep amplitude. II. Description of Data Sources, Accuracy, and Limitations All of the data collected for this project was obtained from the Iowa State University Weather Products webpage. Eight day loops of the 5mb heights and the zonal winds were available at any given time and data needed to be collected before the data was beyond the eight day archiving limit. Data was collected from August 28 th through November 7 th. For this period, wave number, amplitude, and wave speed were collected using the 5mb height maps for both

the Northern and Southern hemispheres. From the zonal wind maps, the 5mb wind speed and 3-15mb average wind speed were collected. a. Waves The first quantity collected was the wave number. This represented the number of waves that crossed the 5 degree latitude in the Northern or Southern hemispheres. A height contour, 558m, was selected and maintained as the standard contour for the Northern Hemisphere for the entire observation period. In the Southern Hemisphere, the height contour of 528m was selected. The wave number was calculated by determining how many pairs of crossings existed between the 558m contour and the 5 degree latitude line. The second quantity was wave amplitude. Amplitude gave a good indication of the significance of a wave. The farther the wave dug towards the tropics or raised towards the poles, the greater the amplitude. A single wave s amplitude was calculated using (1) A = (Z max - Z min ) / 2 where Z max and Z min represent the highest and lowest heights for that wave. Because multiple waves can be present during the analysis of one day s 5mb height field, these needed to be accounted for when determining a final, single value for the wave amplitude. To account for this, the average maxima and minima were calculated to find a representative amplitude for that day. Equation 1 was modified to become (2) A = (AvgZ max + AvgZ min ) / 2 where the terms AvgZ max and AvgZ min were found using AvgZ max = (Z max, 1 + Z max, 2 + Z max, n ) / n AvgZ min = (Z min, 1 + Z min, 2 + Z min, n ) / n The final quantity collected from the 5mb height maps was wave motion, C. This provided an understanding of how quickly the waves were progressing around the hemisphere. It was found by determining how far one of the waves would move over the course of two days and averaging the value to determine an estimate of the speed. (3) C = [LON(day + 1) LON(day 1)] / 2 This was only possible to calculate when the data for a particular day also existed for the previous and following day. b. Zonal Winds Zonal winds were collected through use of another product on the webpage. A global zonal wind average plot was available that gave a cross section of the atmosphere from the South to North Pole, averaging the zonal winds at every latitude around the globe to a single value for that day. The 5 latitude line was used for both hemispheres again in order to find a value for the

5mb zonal wind speed. This allowed for relationships between the waves crossing the 5 latitude and the zonal wind to be analyzed. We also kept track of the maximum wind speed in the 15-3mb layer to determine the strength of the jet stream. c. Accuracy and Limitations Accuracy was a noticeable problem when collecting data for this project. Certain rules needed to be made to determine what was classified as a wave and what was neglected. As mentioned before, a latitude of 5 was selected for both hemispheres as the threshold that needed to be crossed to consider a wave significant. If the target contour for either hemisphere was unable to reach to 5 or only touched it, but did not cross, then it did not meet the criteria for a wave and was neglected in that days data. This means that more waves than represented in the data could have been present on a given day. When it came to analyzing the maps, many difficulties were present due to the low resolution of each map. Without the color shading to indicate where certain contours were, it would have been extremely difficult to determine which contour was associated with a specific height. In many cases, densely packed height contours or continent outlines became jumbled together and difficult to determine if a particularly shallow wave is clearly crossing the 5 or not. The main limitation to determining the correct values for zonal wind was in finding the 5 latitude tick mark. The latitude axis was not labeled, so an estimate needed to be made every time the map was analyzed. This may have led to some inaccurate values that were off by around 5 in either direction. Another issue showed up when strong gradients were present. These gradients became tight enough that it was difficult to determine the correct value at the intersection of the 5mb level and target latitude, and the error in estimating latitude could result in very significant errors. III. Results a. Zonal Wind Zonal winds can be used as a loose proxy for wave speed. If the zonal winds are greater, the speed of atmospheric waves will likely be larger. Very different magnitudes and patterns were observed when comparing the northern and southern hemispheres. In the northern hemisphere, 5mb zonal winds generally remained between and 4 m/s with a slight upward trend throughout the observing period. In the southern hemisphere, values were generally larger, ranging from 1m/s to 5m/s during the period. As the southern hemisphere transitioned from winter to summer, a distinct downward trend in zonal wind speeds was apparent. Figures 1 and 2 illustrate these trends. This makes sense meteorologically, as winter months tend to see a larger temperature gradient in both hemispheres, which typically leads to stronger jet flow aloft. In both hemispheres, the speed of the zonal wind was highly variable, which could be attributed to the development and dissipation of strong troughs and ridges throughout the observing period.

Axis Title 5 hpa Zonal Wind (m/s) 45 5 hpa Zonal Wind vs. Day NH 4 35 3 25 2 15 1 5 19-Aug 8-Sep 28-Sep 18-Oct 7-Nov 27-Nov Date 5 hpa Zonal Wind vs. Day NH Linear (5 hpa Zonal Wind vs. Day NH) y =.25x - 829.53 R² =.27 Figure 1 6 5 hpa Zonal Wind vs. Day SH 5 4 3 2 1 19-Aug 8-Sep 28-Sep 18-Oct 7-Nov 27-Nov Axis Title 5 hpa Zonal Wind vs. Day SH Linear (5 hpa Zonal Wind vs. Day SH) y = -.238x + 8421.6 R² =.215 Figure 2 Since zonal winds can be used as a proxy speed to figure out wave speed, comparing it to the wave number can be a useful way to determine how fast weather conditions change over a certain area. As we can see from figures 3 and 4, no significant correlation can be seen between the two values in either hemisphere. Large wave numbers are often observed in tandem with

5 hpa Zonal Wind (deg/day) 5hPa Zonal Wind (deg/day) large zonal wind speeds, but small wave numbers are seen just as often, so it is difficult to draw any conclusions from this. 6 5 4 5hPa Zonal Wind vs. Wavenumber NH 3 2 5hPa Zonal Wind vs. Wavenumber NH 1 2 4 6 Wavenumber Figure 3 5 hpa Zonal Wind vs. Wavenumber SH 7 6 5 4 3 2 1 2 4 6 8 Wavenumber 5 hpa Zonal Wind vs. Wavenumber SH Figure 4 An interesting feature observed when analyzing zonal wind speeds is their relation to the growth and decay of waves, especially in the northern hemisphere. There are two distinct events,

Wave Amplitude (m) the 8 th of September and the 18 th of October when zonal winds begin to ramp up toward a maximum. At the same time, it appears the wave amplitude begins to increase, which hints at a correlation between increased zonal wind speeds and the deepening of troughs and ridges embedded in the global circulation. This feature is not apparent in the southern hemisphere, so it is difficult to say whether or not there is a physical relationship between these two features, or if they just matched each other by chance in the northern hemisphere. In addition, figure 6 shows that the R 2 value for the northern hemisphere is less than.1. 35 3 25 2 5hPa Zonal Wind and Wave Amplitude vs. Date SH y =.1365x - 5436.3 R² =.56 5 hpa Zonal Wind SH (m/s) Wave Amplitude SH (m) 15 1 5 19-Aug 8-Sep 28-Sep 18-Oct 7-Nov 27-Nov Date Linear (5 hpa Zonal Wind SH (m/s) ) Linear (Wave Amplitude SH (m)) y = -.238x + 8421.6 R² =.215 Figure 5

Wave Amplitude (m) 3 25 2 15 1 5 5hPa Zonal Wind and Amplitude vs. Date NH 19-Aug 8-Sep 28-Sep 18-Oct 7-Nov 27-Nov Date y =.3253x - 13238 R² =.417 5hPa Zonal wind NH (m/s) Wave Amplitude NH (m) Linear (5hPa Zonal wind NH (m/s)) Linear (Wave Amplitude NH (m) ) y =.25x - 829.53 R² =.27 b. Wave Amplitude Figure 6 Wave amplitude was useful for determining how significant the waves were on average for a single day. The quantity represented how large the difference was between the average maximum and minimum height within the waves that were being classified as a wave in the dataset. In both hemispheres, an increase in the average wave amplitude was observed, but the increase in the Northern Hemisphere was more significant. From Figure 7, the Northern Hemisphere saw an increase of over 2m in average wave amplitude. The daily trend was to increase by about.325m, which would result in an increase of nearly 1m per month. On a time scale of about a week there would be fluctuations that deviate as much as 9m from the average, but the amplitude would only peak at these extreme values and quickly return to the average. No deviation from the trend line was longer than 11 days.

Wave Amplitude (m) 3 Wave Amplitude vs. Date NH 25 2 15 1 5 Date y =.3253x - 13238 Figure 7 The Southern Hemisphere saw a similar trend in the amplitude over the course of the observational period, but it wasn t as significant. As shown in Figure 8, the trend was for the wave amplitude to increase by.136m per day which is only 4% the daily increase seen in the Northern Hemisphere. This led to an increase of about 4m per month, and a total increase of about 1m over the course of the period. There was a period from September 22 nd to October 6 th when the amplitude stayed about 5m above average. Other than that spike, the wave amplitude seemed to fluctuate on an order of two to four days. Comparing the wave number and wave amplitude led to some interesting results. In the Northern Hemisphere, the trend indicated that an increase in wave number resulted in an increase in the wave amplitude. The opposite was found in the Southern Hemisphere, where an increase in the wave number led to a decrease in the wave amplitude. According to Rossby Wave Theory, an increase in the wave number should result in a decrease in the wave amplitude. Only the Southern Hemisphere followed this theory. The Northern Hemisphere violated the theory, but only just.

Wave Amplitude (m) 35 Wave Amplitude vs. Day SH 3 25 2 15 1 5 Date y =.1365x - 5436.3 Figure 8 The Northern Hemisphere trend does indicate that the wave amplitude increases as the wave number increases. From Figure 9, the slope was only +2.4m in amplitude for each additional wave. The maps we used to calculate wave amplitude did come with quite a bit of uncertainty in determining specific values for wave amplitude. The contours were 6m apart which meant estimations needed to be made on specific amplitude size, which can very easily lead to large errors. There also isn t observed data for every single point in the atmosphere, so the maps themselves have a degree of estimation simply in generating the map, so there are possibilities that the amplitudes are slightly inaccurate to real life. Finally, considering the fact that the trend is only an estimation of the real relationship, based on the high amplitude at low wave numbers and low amplitudes at high numbers, a decreasing trend could be drawn. Another indication that there might be an error in the observed data is that it suggests conservation of energy is not entirely present. At a wave number of two, a range of 11m is present in the amplitude. As the wave number increases to three, the range decreases to 12m. This suggests that as the wave number increased, so did the wave amplitude. The trend does follow conservation of energy from wave number of three to five though, and this one instance of disconnect is not enough to declare an error in the data or a violation of conservation. Our data is not conclusive evidence that the Rossby Wave Theory is being broken in the Northern Hemisphere and would require more accurate sampling in order to determine what is truly going on.

Wave Amplitude (m) 3 Wave Amplitude vs. Wave Number NH 25 2 15 1 5 1 2 3 4 5 6 Wavenumber y = 2.418x + 15.27 R² =.81 Figure 9 The Southern Hemisphere did show a trend that matched the Rossby Wave Theory. For every additional wave, the wave amplitude decreased by 11m. Thinking through the lens of conservation of energy, it makes sense that more waves should result in less amplitude. If more waves also had higher amplitude, that would suggest that even more energy is present in the system. From one wave to another the range in amplitudes shifts down and becomes tighter, leaving very little evidence that the trend should be increasing like in the opposite hemisphere. Another thing to note is the variability in the range of amplitudes for each wave number. The range decreased from 16m at a wave number of two to 5m at a wave number of five. Only one data point is present at a wave number of six and can t be used to determine if the trend continues, but logically it should continue to narrow. This goes back to the discussion of conservation of energy. As more and more waves are present, they must, on average, decrease in amplitude in order to conserve the total energy of the system.

Wave Amplitude 35 Wave Amplitude vs. Wave Number SH 3 25 2 15 1 5 1 2 3 4 5 6 7 Wave Number y = -11.93x + 226.44 R² =.882 Figure 1 c. Wave Number The wave numbers for each hemisphere had distinctly different patterns. In the Northern Hemisphere, the wave number steadily increased from one to four. In the Southern Hemisphere, the wave number slowly decreased from four to three. Both of these trends can be explained through the change of the seasons. In the Northern Hemisphere, the change to fall meant the heights falling much more than in the summer. This made it more likely for the 558m contour to reach 5 N. In the Southern Hemisphere, the transition from winter to late spring made it less likely for the 552m contour to cross the 5 S because warmer temperatures are helping to raise the heights over the hemisphere. Looking at Figure 11, a linear trend line shows that the wave number increases steadily by approximately.388 per day. This means it will take roughly 26 days to increase the number of waves by one. That corresponds with around a one wave increase per month in the Northern Hemisphere. The average number of waves for the Northern Hemisphere was 3.1, and for the course of about September 1 th to October 3 th, the Northern Hemisphere was within one wave of that average. It s worth noting that statistics don t accurately represent what actually happened. For this same period of time, the wave number was often fluctuating between one and five waves, which are +/- 2 waves from the average. During two periods, each lasting one week, the wave number remained constant at four. During the period from October 3 rd to October 28 th, the wave number remained with +/- 1 of a wave number of four. At an earlier date, from August 29 th to September 18 th, the wave number remained with +/- 1 of two. This shows an obvious change in the average wave number between the summer and fall seasons. The Northern

Wave Number Hemisphere saw a lot more variability in the wave number during the transition from summer to late fall than the Southern Hemisphere. 6 5 4 3 2 1 Northern Hemisphere Wave Number Date y =.388x - 1594.9 Figure 11 Figure 12 shows the wave number for the Southern Hemisphere and also displays the decreasing linear trend over the course of the observational period. The trend for the Southern Hemisphere was to decrease by.91 per day. This means it would take roughly three and a half months for the wave number to decrease by one. Our observational period was only 71 days and cannot confirm if that trend would hold true. The average wave number was 3.6 which is larger compared to the Northern Hemisphere. The Southern Hemisphere was more stable over the course of the period and never saw the wave number decrease to one. From August 29 th to September 16 th the wave number remained with +/- 1 of four. In a later period, from October 11 th to November 1 st the wave number was within +/- 1 of three. As mentioned before, this change is not as significant as in the Northern Hemisphere, but it does follow the opposite trend as expected.

Wave Number Southern Hemisphere Wave Number 7 6 5 4 3 2 1 Date y = -.91x + 376.51 Figure 12 d. Wave Speed Wave patterns move at speeds that vary from season to season and even slightly from year to year. Figure 13 below shows the wave motion in both the northern and southern hemispheres at 5 degrees latitude. On average for both hemispheres, waves travel around the 5 degree latitude circle at just under 15 degrees per day. That would equate to just under 4 weeks for a wave to fully circle the 5 degree parallel. Another thing to note is that all of the recorded values for wave speed are positive, which means that waves at 5 degrees latitude in both hemispheres travel from west to east. This agrees with Rossby Wave Theory for sufficiently positive zonal wind values. The positive zonal wind values indicate that the prevailing westerlies are helping to move the waves from west to east. We will look more into the wave speed/zonal wind relationship later. As one would expect, the values for wave speed are higher in the southern hemisphere during the months of August and September, and the wave speeds even out for both hemispheres as the fall season approaches in the Northern hemisphere.

Figure 13 Figure 14 and 15 are plots of the 5hPa wave speed versus the 5hPa zonal wind for the northern and southern hemispheres respectively. It is important to note that the values for both of these parameters have been converted to degrees per day in order to make comparisons. Averaging the 5 hpa wave speed for the northern hemisphere, we get +11.3 degrees/day, and we get a slightly higher number of +17.3 degrees/day when averaging out the 5 hpa zonal winds for the northern hemisphere. This tendency of the zonal wind values to be greater than the 5hPa wave speed numbers is also true to a greater extent in the southern hemisphere, where the values for the 5 hpa wave speed and zonal wind are +15.8 and +34.5 degrees/day, respectively. Even though the average difference between the 5 hpa zonal wind and the wave speed is greater in the southern hemisphere, the slope of the linear regression line that explains the variation of wave speed versus zonal wind is larger for the northern hemisphere, as depicted by figure 14. This all makes good sense in terms of Rossby Wave Theory as the equation for phase speed, c x, is illustrated by c x =u-{β/(k 2 +l 2 )}. The β-term, which is the N-S advection of planetary vorticity, will try to make the waves propagate westward, which would make the wave speed more negative. Thus, the zonal wind speed, u, must be greater than the wave speed in order for this Rossby Wave Theory equation to make sense.

Wave Speed (deg/day) Wave Speed (deg/day) Wave Speed vs. 5hPa Zonal Wind NH 25 2 15 1 Wave Speed vs. 5hPa Zonal Wind NH Linear (Wave Speed vs. 5hPa Zonal Wind NH) 5 1 2 3 4 5 6 U5 Zonal Wind (deg/day) y =.858x + 9.8498 R² =.71 Figure 14 Wave Speed vs. U5 Zonal Wind SH 35 3 25 2 15 1 5 Wave Speed vs. U5 Zonal Wind SH Linear (Wave Speed vs. U5 Zonal Wind SH) y =.296x + 14.753 R² =.33 2 4 6 8 U5 Zonal Wind (deg/day) Figure 15

Wave Speed (deg/day) Looking at the 5 hpa wave speed versus the zonal winds at jet stream level for both hemispheres, we can see that there is a weak positive correlation between these two parameters. Essentially, as the jets stream wind increases, the wave speed increases. Even though we only have weak correlation coefficients of.4 and.8 for the northern and southern hemispheres, respectively, the positive values of the slope of the linear regression lines for both hemispheres do agree with Rossby Wave Theory. As stated above, as the zonal winds become more positive, Rossby Wave Theory would argue that the wave speed will also become more positive. I summary, Rossby Wave Theory does agree with the zonal wind data at both the 5 hpa level and at the jet stream level. 25 Wave Speed vs. U15-3hPa Zonal Wind NH 2 15 1 5 Wave Speed vs. U15-3hPa Zonal Wind NH Linear (Wave Speed vs. U15-3hPa Zonal Wind NH) 2 4 6 8 U15-3hPa Zonal Wind (deg/day) y =.45x + 9.8939 R² =.412 Figure 16

Wave Speed (deg/day) Wave Speed vs. 15-3hPa Zonal Wind SH 35 3 25 2 15 1 Wave Speed vs. 15-3hPa Zonal Wind SH Linear (Wave Speed vs. 15-3hPa Zonal Wind SH) 5 2 4 6 8 1 12 Zonal Wind 15-3hPa (deg/day) y =.174x + 9.9542 R² =.81 Figure 17 Now we will plot the difference between wave speed and 5 hpa zonal wind versus the integer wavenumber for both the northern and southern hemispheres. Figures 18 and 19 show this respectively. The difference seems to decrease as the integer wavenumber increases in the northern hemisphere, while in the southern hemisphere, there really is no clear relationship between these two parameters. According to Rossby Wave Theory, the phase speed, c x, is equal to the frequency of the wave divided by the wavenumber. Wave speed should increase as the wavenumber decreases, which means that the difference between the wave speed and the 5 hpa zonal wind should also increase as the wavenumber decreases. This is not showing up in either of the tow hemispheres, so these two figures below disagree with Rossby Wave Theory. This disagreement may be true either due to measurement error when acquiring the data from the Iowa State Weather Products website, or it may be true because to the assumptions we make when using Rossby Wave Theory, such as a barotropic atmosphere. Barotropic means that all levels of the atmosphere have the same values for pressure and density, which we know is not exactly true. More research could be done on this topic to correctly determine the probable causes of this error.

Wave Speed-U5 Zonal Wind (deg/day) Wave Motion-U5 Zonal Wind (deg/day) -.5 Wave Speed-5hPa Zonal Wind (deg/day) NH 1 2 3 4 5 6-1 -1.5-2 -2.5 Wave Speed-5hPa Zonal Wind (deg/day) NH -3-3.5-4 -4.5 Wavenumber Figure 18 Wave Speed-5hPa Zonal Wind (deg/day) SH 2 4 6 8-1 -2-3 -4 Wave Speed-5hPa Zonal Wind (deg/day) SH -5-6 -7 Wavenumber Figure 19

e. 5hPa Heights vs. 5hPa Zonal Winds Now we will try to apply the data we recorded to a couple of real world situations. Figure 2 shows the 5hPa heights plotted for Ames, IA versus the 5hPa northern hemisphere maximum zonal wind from 9/16/12 to 9/26/12. It is interesting to note that most of the highest zonal wind values at 5 degrees north occurred when the 5hPa heights in Ames were highest. This is probably true because during the late summertime the main jet stream would be located north of Ames at 5 degrees latitude when 5hPa heights in Ames are high. If the heights in Ames are lower, however, the 5hPa jet would be located closer to Ames, thus the 5hPa zonal winds at 5 degrees north would decrease as the jet moves south. Another event that just occurred recently was Hurricane Sandy. Figure 21 shows 5mb height for New York City (NYC) versus the 5hPa zonal wind over NYC from 1/25/12 to 1/31/12. There is a clear negative correlation between these two parameters. As the 5hPa heights fell to around 54 decameters over NYC due to Hurricane Sandy being in extratropical transition, the zonal winds decreased rapidly. As we can see, it is probably reasonable to assume that when a location is experiencing cyclonic flow, zonal winds will increase as 5hPa heights decrease, on average. More research needs to be done and more cases need to be looked at on this, but a clear relationship was observed during Hurricane Sandy over NYC. Figure 2

Figure 21 IV. Conclusion We have seen that some of the results fit well and agreed with Rossby Wave Theory, while others did not. We also should note that with the exception of the graph about Hurricane Sandy, R 2 values did not go much above.1. The mathematical interpretation of this is that less than 1 percent of the variability in our graphs can be explained by the linear regression lines. Perhaps, we would be able to get higher R 2 values if we recorded data for a longer period of time, or if we did not make all of the assumptions of Rossby Wave Theory. In other words, we would assume a more realistic, non-barotropic atmosphere. More research could be done on this and eventually, we may be able to assist operational meteorologist in forecasting the 5hPa wave patterns throughout the world.