Tracking of Large-Scale Wave Motions Nikki Barbee, Adam Cale, Justin Wittrock Dr. William Gutowski Meteorology 44 Fall 29 This semester we have observed large scale wave patterns in both the Northern and Southern Hemispheres. We took data including wavenumber, amplitude, wave speed, and maximum zonal winds at both hpa and in the upper atmosphere. The goal of this data collection is to look at equations in our book and test the Rossby wave theory to see if it applies to large scale waves in our atmosphere. In this paper we hope to answer the following questions: how rapidly do wave patterns move, is there a relationship between zonal winds and wave propagation, how long does one identifiable pattern last, how rapidly do waves change the amplitude, how do zonal winds evolve, and is there a relationship between zonal winds and wave growth or decay. First we will look at how fast wave patterns move and the relationship between zonal wind maxima and the wave pattern. 1
The Speed of the Wave Patterns To determine the speed that the wave patterns move and the relationship between zonal wind speeds and wave propagation, we need to look at the very specific parameters, the first of which is the overall average wave speed in degrees per day. To do this, we add up the wave speeds and divide the sum by the total number of days we observed. Our average for the Northern Hemisphere was 1.8 o per day. Likewise, the average for the Southern Hemisphere was 1.3 o per day. This means that on average the waves in the Southern Hemisphere traveled nearly o per day faster than those in the Northern Hemisphere. The waves are moving eastward in both hemispheres. This means counterclockwise rotation around the North Pole and clockwise rotation around the South Pole. Knowing these speeds we can calculate how long it would take an intact wave to move around the globe in each hemisphere. To do this, we divide 36 o by the speed in degrees per day. For example in the Northern Hemisphere we obtain (36 o ) / (1.8 o /day) = 33.3 days. Similarly, for the Southern Hemisphere we obtain 23. days. This makes sense because the waves are traveling faster in the Southern Hemisphere. Now we can look at how wave speed varies with zonal wind particularly at hpa. Relationship between Zonal Wind Speed and Wave Propagation First we need to convert wind speed into degrees per day. Then taking the length of the latitude circle at o and dividing it by wind speed tells us how many days it would take to travel 36 o at this speed. So for the Northern Hemisphere when converting the average wind to degrees per day we calculate the following equation: [(9.26 m/s)(36 s/hr)(24 hr/day)] / [(111 km/1 o )(1 m/km)]. This gives us an average wind speed of 7.2 o per 2
Wave Speed ( o /day) day instead of it being in m/s. Working this equation for the Southern Hemisphere, we get a value of 19.3 o per day. Dividing 36 o by these speeds gives us the number of days it would take to go around the globe. For the Northern hemisphere we have (36 o ) / (7.2 o /day), which give us an answer of days. For the Southern Hemisphere we get 18.7 days. Looking at Figure 1, we see the wave speed plotted against the U- speeds in degrees per day for the Northern Hemisphere. Correspondingly, Figure 2 shows the same data for the Southern Hemisphere. At first it seems odd that the wind speeds work out to be in multiples of four; however, when you work out the conversions to degrees per day from meters per second that is what you end up with: 2 Northern Hemisphere 2 1 1 Speed (o/day) 1 1 2 U- ( o /day) Figure 1: Wave Speed vs. hpa Wind Speed for the Northern Hemisphere 3
Wave Speed ( o /day) 3 Southern Hemisphere 2 2 1 1 Speed (o/day) 1 2 3 4 U- ( o /day) Figure 2: Wave Speed vs. hpa Wind Speed for the Southern Hemisphere Knowing what we know about the Rossby waves and our data, Ubar would be positive as well as the beta term. This means our plot for the Northern Hemisphere makes sense for the faster wind speeds but not the lower ones. Since both terms are positive the speed of a Rossby wave should be less than U-bar. This isn t what we see with the lower wind values in the Northern Hemisphere. Figure 2 shows the same plot for the Southern Hemisphere, and we see that the plot agrees with the Rossby wave theory much more so than the Northern Hemisphere s did. Figures 3 and 4 respectively show the upper level wind speeds versus the wave speed both in degrees per day. 4
Wave Speed ( o /day) Wave Speed ( o /day) 2 Northern Hemisphere 2 1 1 Speed (o/day) 1 2 3 4 U-upper ( o /day) Figure 3: Wave Speed vs. Upper Level Wind Speed in the Northern Hemisphere 3 Southern Hemisphere 2 2 1 1 Speed (o/day) 1 2 3 4 6 U-upper ( o /day) Figure 4: Wave Speed vs. Upper Level Wind Speed for in the Southern Hemisphere Along with these plots we can solve the time it would take to go around the globe just as we have done at the hpa level. These values are 21.8 days for the Northern Hemisphere and 11. days for the Southern Hemisphere. These plots show that there is a
Wave Speed ( o /day) certain dependence on upper air speeds by the wave speed, which demonstrates the same relationship that we saw in the hpa. The Rossby wave speed is higher than the actual wave speed for both hemispheres. This fits much better for the northern hemisphere than the hpa plot did. The Rossby theory also shows a relationship between wave speed and wavenumber Figures and 6 display that relationship for the Northern Hemisphere and the Southern Hemisphere respectively. 2 Northern Hemisphere 1 1 - c - U- (o/day) Poly. (c - U- (o/day)) -1 2 4 6 8 Integer Wavenumber Figure : Wave Speed vs. Wavenumber in the Northern Hemisphere 6
Wave Speed ( o /day) Southern Hemisphere 2 1 1 - -1-1 -2-2 -3 2 4 6 c - U- (o/day) Poly. (c - U- (o/day)) Integer Wavenumber Figure 6: Wave Speed vs. Wavenumber in the Southern Hemisphere The correlation we see from Figure and 6 is a squared relationship. However, the interesting thing is that in the Northern Hemisphere the trend is negative while in the Southern Hemisphere it is positive. We are not sure why this occurs; it might just be the different hemispheres. The next question we will address is how long identifiable patterns last. The Longevity of a Pattern We have patterns that last up to two weeks with wavenumbers varying by only one integer value. You can see this in Figure 7 for the Northern Hemisphere and Figure 8 for the Southern Hemisphere. This time frame is in alignment with typical synoptic scale events. The longest time frame we had with no wavenumber oscillation was 6 days in the Southern Hemisphere. 7
Integer Wavenumber Integer Wavenumber 7 Northern Hemisphere 6 4 3 2 Wavenumber 1 Date Figure 7: Wavenumber vs. Date for the Northern Hemisphere 6 Southern Hemisphere 4 3 2 Wavenumber 1 Date Figure 8: Wavenumber vs. Date for the Southern Hemisphere 8
Amplitude Looking at the amplitude of the waves vs. wavenumber we see that the shorter waves tend to have smaller amplitudes. In Meteorology 44 class we discussed the following expected relationship: the shorter waves have higher amplitude than the longer waves. Unfortunately, our observations showed the opposite. This relationship is shown in Figure 9 for the Northern Hemisphere and Figure 1 for the Southern Hemisphere. This might have something to do with the mb wind speed affecting the wave amplitude. Northern Hemisphere 4 3 3 2 2 1 1 2 4 6 8 Integer Wavenumber Amplitude (m) Figure 9: Amplitude vs. Wavenumber for the Northern Hemisphere 9
Amplitude 4 3 3 2 2 1 1 Southern Hemisphere 1 2 3 4 6 Integer Wavenumber Amplitude (m) Figure 1: Amplitude vs. Wavenumber for the Southern Hemisphere The next question we will address is how rapidly waves increase or decrease in amplitude. The Rise and Fall in the Amplitude of the Waves There are times when we can mark grown and decay. The typical time frame for these in the Northern Hemisphere is 1 days. In the Southern Hemisphere the growth lasts about two weeks. In the Northern Hemisphere the largest growth was about 3% in ten days. The Southern Hemisphere had two periods of marked growth. One was about 3% increase over fourteen days and the other was about 2% over the span of about 17 days. A plot of Amplitude vs. Date is shown in Figure 11 for the Northern Hemisphere and Figure 12 for the Southern Hemisphere: 1
Amplitude Amplitude 4 3 3 2 2 1 1 Northern Hemisphere Amplitude (m) Date Figure 11: Amplitude vs. Date for the Northern Hemisphere 4 3 3 2 2 1 1 Southern Hemisphere Amplitude (m) Date Figure 12: Amplitude vs. Date for the Southern Hemisphere 11
U- ( o /day) Overall Zonal Wind Trends Looking at the zonal winds throughout the entire observation period we see that the overall trend in the Northern Hemisphere is slightly increasing in term of wind speeds over the period; this is shown in Figure 13. The overall trend in the Southern Hemisphere is decreasing in term of wind speeds as the period progressed; this is shown in Figure 14. The trend for the Southern Hemisphere is much more pronounced than that of the Northern Hemisphere. This could be due to the seasonal shift from winter to spring being more extreme than that of summer to fall. 18 16 14 12 1 8 6 4 2 Northern Hemisphere U- (o/day) Linear (U- (o/day)) Date Figure 13: hpa Windspeed vs. Date 12
U- ( o /day) 4 4 3 3 2 2 1 1 Southern Hemisphere U- (o/day) Linear (U- (o/day)) Date Figure 14: hpa Windspeed vs. Date for the Southern Hemisphere Relationship between Zonal Wind and Wave Growth or Decay Looking back at Figures 11, 12, 13, and 14 we can compare the zonal winds and the amplitudes on any given day. The trend that is most noticeable is that whenever the wind speeds are higher, the amplitude tends to be higher. This occurs in both hemispheres. This would be expected because the winds can dig the troughs deeper and make the ridges higher. Conclusion Overall our observations show some discrepancies with the Rossby theory. The one that sticks out the most is that our longer wavelengths had larger amplitudes than our shorter wavelengths. One thing that did agree was that our longer waves did have higher wind speeds in general. Because our data is recorded by human hands, there are potential 13
recording errors in our data. Therefore, our results may not be spot on. All in all though, this project answers most of our questions regarding the wave analysis. 14