Analysis of 500 hpa Waves for the Northern and Southern Hemisphere

Transcription

1 Analysis of hpa Waves for the Northern and Southern David Dziubanski, Mike Greve, and Rachael Witter Department of Geological and Atmospheric Sciences Iowa State University, Ames, IA Abstract This study examined hpa waves and how well their behavior corresponds to Rossby wave theory. Analysis was done on wave number, amplitude and wave propagation speed, as well as average zonal wind at hpa and the layer of 3-1 hpa. Findings indicate that this analysis was somewhat inconclusive due to low correlation coefficients. Errors in the analysis and data may have played vital in the findings of this report. Some aspects of this study agree with Rossby wave theory, while others did not. 1. Introduction Knowing how hpa waves move and change over time can be useful for many different things such as forecasting or studying climate over short periods of time. Rossby wave theory indicates a relationship between zonal wind speed and wave speed. A study of hpa waves and the Rossby wave theory is presented here. Over a span of eight weeks, the integral wave number, amplitude, wave speed, hpa maximum wind speed, and 3-1 hpa wind speed were recorded and analyzed to find averages and trends in the data to study the wave patterns and if Rossby wave theory is able to be applied to waves in the atmosphere for real time data. 2. Data and Methods Waves from the -hpa field were observed over the period from September 1 th, 21 through November 1 th, 21 over the Northern and Southern hemispheres. Rossby waves from this field were examined to find features that described the waves shape and motion. Wave number, amplitude of Rossby waves and wave speed were determined by observing these waves

2 through the period. Wave number was found by using a target contour of 8 meters, and a target latitude circle of N for the Northern hemisphere. This value was found by counting the number of times this target contour crossed the latitude circle. For the Southern hemisphere, a similar process was done utilizing the 28 meter contour and the S latitude circle. Amplitude was calculated using the deviation from these target contours of the Northern and Southern hemispheres, respectively. Wave propagation speed was determined by observing the motion of waves along the latitude circle, in degrees longitude per day. In addition to observations of the Rossby waves, the average zonal wind was also examined. The zonal wind refers to the average u-component of the wind around a latitude circle. This quantity was examined at several pressure levels of the atmosphere. Maximum - hpa wind speeds as well as maximum wind speeds in the upper levels of the atmosphere (3 hpa 1 hpa) were observed. Wind speed and pressure level of this occurrence were recorded. It is clear that errors in data observation may occur. These errors may arise from data collection due to lack of exactness in measuring wave speeds and amplitudes. From the zonal wind plot, errors in determining exact latitude of the maximum and minimum wind speeds could also arise due to the scale of the latitude axis. 3. Analysis How rapidly do wave patterns move? The Northern Motion is mostly confined to phase speeds between -1 degrees per day (Fig 1). In the fall season when the NH hemisphere was transitioning from summer to fall, especially in September, the motion was highly variable, ranging from speeds of. to 2 degrees per day. This may be due to the larger temperature contrasts setting up between northern and mid latitudes. These temperature contrasts would cause larger trough and ridge amplification in different regions and mid latitude cyclone development. Both of these factors could affect the phase speed. However, as October progressed, the motion became less variable and slower between to 1 degrees per day. This may be partly due to the blocking that was taking place in October between 6E and 6W during this time frame. This blocking would essentially slow down motion by interrupting the normal zonal flow. The Southern motion is continuously variable through the entire period, mostly ranging between and 1 degrees per day (Fig 2). The period from the middle of September through early November was characterized by an increase in wave motion, as is depicted by the trend line. This steady increase and variability is most likely due to the transition period occurring in the Southern at this time. Developing storm systems due to the remnant cold air moving off of Antarctica most likely can account for the strong variability in motion. The steady increase in phase speed might be due to the increase in amplitude that occurred over this time period as the Southern transitioned from winter to summer.

3 Is there a relationship between zonal wind speed and wave propagation? Rossby wave theory suggests a distinct relationship between zonal wind speed and wave speed. Comparisons are made in this section to find if the data fits the Rossby wave theory. A look at the average wave speed and direction of waves is done first, then a comparison of wave speed to hpa wind speed is done, also a comparison of wave speed to the maximum wind speed in the layer of 3 to 1 hpa is done, and finally, a comparison of wave speed without a background zonal wind speed is compared to wave number. hpa waves in both the North and South hemispheres had a general west to east movement. On average the wave speed in the Northern hemisphere was 8.4 deg/day and in the Southern hemisphere, 11.8 deg/day. These numbers indicate a faster wave movement in the Southern hemisphere. A wave would take around 43 days to move around a latitude circle in the Northern hemisphere while taking only 3. days in the Southern hemisphere. In the Northern hemisphere the wave speed shows an increase with larger hpa U speeds and the same is true for the Southern hemisphere (Fig. 3 and 4). The wind speed at hpa for the Northern hemisphere was 19.3 deg/day and for the Southern hemisphere, 29. deg/day. Both of these values are larger than their respective average wave speeds. This and the fact that both hemispheres showed increasing trend lines for wave speed compared to the U indicate a compliance with the Rossby wave theory. The theory indicates that the wave speed is proportional to the zonal wind speed from the dispersion relation, where c = wave speed, ū = zonal wind average, = the change in absolute vorticity in the meridional direction, and k and l are both horizontal wave numbers. This relation indicates that as the zonal wind increases, so too should the wave speed. It also explains the reason why the zonal wind average is larger than the wave speed and according to Rossby wave theory; it should be by a factor of.the data is much too variable for an accurate portrayal of real results, but indications from increasing trends and larger U values agree with Rossby s wave theory. The graphs seen in Figs. (NH) and 6(SH) show wave speed compared to the upper layer wind speed. The trends seen are slight increases for both hemispheres but the slopes for the equations were fairly small and the data was too variable as well with rather low R² values. This possibly indicates that the relationship between upper layer 3-1 hpa winds to the wave speed might not be as important as the U winds to the wave speed. The final comparison in this section was done by removing the dependence of the wave speed on the U by comparing (wave speed - U) with wave number. For the Northern hemisphere (Fig. 7), the wave speed alone tends to increase very slightly with larger wave number and the opposite is true for the Southern hemisphere (Fig. 8). The relationship is consistent with Rossby wave theory in the Northern hemisphere while the Southern hemisphere rejects the theory. Rossby wave theory states that as the integral wave numbers become larger, the wavelengths become shorter. If the wavelengths are shorter, the waves tend to move faster and the opposite is true for longer wavelengths. This again is related back to the dispersion relation equation, where the relationships for different variables can be seen. (1)

4 How long does one identifiable pattern last? This section will cover the relationship between wave numbers and time as well as wave number and amplitude. The amplitude of a wave can be used to discover long and short waves as well as heights of waves. The relationship between wave numbers and time can indicate how closely the wave pattern is comparing to the synoptic scale and what kind of weather is affecting the area. For the Northern hemisphere a wave number of 4 ( seen in Fig. 9) is the most dominant wave number and the interesting result seen from the data analysis is that this wave number is very dominant from early October to early November. The only variance is + 1 and this is not considered to be significant enough to cause a shift in the wave pattern. The synoptic scale discussed by Rossby s wave theory suggests a scale of close to 1 day, but for the Northern hemisphere in this research, the scale appeared to be close to a month. Possible reasons for this rather large scale might include the strong blocking scheme in this month which kept weather fairly quiet for most of October over much of the United States. This omega blocking scheme could have had something to do with the dominant wave number 4 for this time period. The Southern hemisphere exhibited more a synoptic scale pattern. The dominant wave number was three and seemed to show a three to four day pattern where it would go up to six and down to two or even one at one point towards the end of the period as seen in Fig. 1. The three to four day pattern is still longer than the one day Rossby wave theory pattern, but much smaller than the Northern pattern. When amplitude was plotted versus wave number (Fig. 11 for NH and Fig 12 for SH), the longer waves tended to have larger amplitudes than did the short waves for both hemispheres. This result would reject the wave the Rossby wave theory, which indicates short waves, should have the higher amplitude because they move faster and are able to become deeper because of the faster movement. Error in analysis is most likely the cause in the rejection of the theory in this section. How rapidly do waves increase or decrease in amplitude? Amplitude vs. is plotted for both hemispheres in Figures 13 and 14, respectively. Overall, the Northern hemisphere shows a slight linear increase in amplitude as the period progresses. Conversely, the Southern hemisphere shows a linear decrease in amplitude. The Northern hemisphere shows a slight increase at the beginning of the period, starting at an amplitude of 146. meters. This trend decreased for approximately two days. The amplitude decreased rapidly, reaching a minimum of meters, in the last week of September until the first week of October. The pattern took another increasing trend from early October until mid October, reaching a local maximum of 227. meters on October 17 th. After this increasing trend, the pattern dropped off and remained almost steady during the last week of October. The pattern then increased once again, to a maximum amplitude of 2 meters on November 4 th. A decreasing trend continued through the rest of the data set, with a very slight increase over the last few days of observing.

5 The Southern hemisphere exhibited a similar pattern, but decreasing. The amplitude started out as 3 meters in early September, and then decreased until early October. A minimum of 12 meters was shown on October 1 st. After this point, the pattern showed a sharp increase until the middle of October. A local maximum of 322. meters was observed on October 13 th. The pattern showed steady amplitudes ranging from 16 meters to 2 meters from October 19 th through October 29 th. After this point, a slight increase was shown until November 3 rd, when a maximum of 28 meters was observed. The trend decreased for the rest of the data set, reaching a minimum of 13 meters on November 1 th. How does zonal wind evolve through the period? Maximum -hpa Wind Speed vs. is plotted on Figures 1 and 16. The Northern hemisphere showed a linear increase trend as the period progressed. A minimum value of - hpa wind speed, meters per second, was observed on September 1 th. This wind speed increased to 1 meters per second by September 17 th and remained steady for four days. Slight decreases and increases are exhibited until the end of September and into early October, where a sharp increase to 2 meters per second was observed on October 6 th. This also decreased sharply once the maximum was reached, to 1 meters per second on October 8 th. The wind speed remained steady at 2 meters per second for the second week of October, the decreased the following week, reaching a minimum of 1 meters per second for a duration of three days. Another sharp increase to 3 meters per second was observed on October 23 rd. Wind speeds held steady at 2 meters per second over a duration of 4 days following this increase. Overall, the trend was decreasing until approximately November 1 th, and then was followed by a sharp increase, reaching the maximum wind speed observed of 4 meters per second on November 14 th. The Southern hemisphere also showed a linear increase, however not as drastic as the Northern hemisphere. A maximum of 3 meters per second was observed on September 1 th. This trend was generally decreasing until reaching a minimum of 1 meters per second on September 2 th. This quickly increased to a maximum of 4 meters per second just two days later. After this increase, a decrease in maximum wind speed is exhibited. A second minimum of 1 meters per second was observed on October 3 rd. The -hpa wind speed showed a steady increase to 3 meters per second by October 16 th. After this date, the pattern fluctuates between 2 meters per second and 3 meters per second until November 2 nd, when another decrease is observed. This decrease is followed by more fluctuations, this time between 2 meters per second and 1 meters per second, until sharply increasing to the maximum of the period, meters per second, on November 1 th. Is there a relationship between zonal wind speed and wave growth/decay? Over the period between mid September and mid November, U and Uupper increased greatly in strength across the Northern (Fig. 17). At the beginning of the period,

6 U was typically around 1 m/s average, and Uupper was stronger with wind speeds near 2 m/s average. By the beginning over November, U increase to around 2 m/s average, and Uupper increased to 3-4 m/s average. This increase in wind speed can be attributed to the transitioning weather across the NH. The amplitude increased on average from 14 m in September to 18 m in late October. Both the zonal wind and amplitude have a high correlation. As the zonal winds increased, the amplitude increased. Larger amplitudes due to large air mass differences typically create stronger jet streams. These zonal winds represent the increased jet stream strength due to larger amplification. The Southern hpa and Upper level winds both have a similar trend from September through November (Fig. 18). U increases from 2 m/s to 3 m/s average between late September and late October. The wind strength then decreases in early November. Uupper increases from 3 m/s to 4 m/s average between late September and early October. Uupper then maintains the average 4 m/s strength through the month of October before decreasing in early November. The amplitude has a high correlation with the zonal wind strength. The amplitude increases in late September from near 16 m to 2 m in early October. The amplitude then remains fairly large through the entire month of October, with a couple large amplitude peaks over 2 m. This increase in amplitude and zonal wind signifies the transition months in the Southern. Remaining cold air interacting with the warm air building to the north creates stronger jet streams and mid latitude cyclones, which translates to large amplitude. One factor to note is the lag between the increase in zonal wind and the increase in amplitude. The response of the amplitude might take longer to evolve than the zonal wind, which could change rapidly on short time scales. 4. Conclusion This study looked at hpa waves in both the Northern and Southern hemispheres. Data was taken over a time span of 8 weeks and analyzed for integral wave number, amplitude, wave speed, and hpa maximum wind speed along with 3-1 hpa maximum wind speed. All of these values were put into Excel 27 and were then compared and graphed to find trends and averages. Analysis of these values led to some agreement with the Rossby wave theory, but mostly disagreement. The Northern showed agreement when wave speed alone was compared to wave number, but all other aspects failed in some way or another. hpa wind speeds did somewhat fit with 3-1 hpa wind speeds but did not show excellent compliance as well as the linear relationship between amplitude and wave number, where the wave theory again failed. Possible errors in the data collection from color height lines to interpretations on where the latitude line was on the zonal wind chart could have led to the rejections of the wave theory along with other errors such as only using one latitude circle to represent the theory. Further analysis might try and remove human errors as well as using multiple latitudes to test Rossby s wave theory.

7 Wave Speed(deg/day) Wave Speed (deg/day) 3 Wave Speed vs., Northern Fig. 1, Wave Speed vs., Northern Wave Speed vs., Southern Sep 22-Sep 29-Sep 6-Oct 13-Oct 2-Oct 27-Oct 3-Nov 1-Nov Fig. 2, Wave Speed vs., Southern

8 Wave Speed (deg/day) Wave speed (deg/day) 3 2 hpa zonal wind speed vs Wave Speed NH y =.278x hpa zonal wind (m/s) R² =.33 Fig. 3, hpa Zonal Wind Speed vs. Wave Speed, Northern 3 hpa zonal wind speed vs. Wave speed, Southern y =.119x hpa zonal wind (m/s) R² =.88 Fig. 4, hpa Zonal Wind Speed vs. Wave Speed, Southern

9 Wave Speed (deg/day) Wave Speed (deg/day) 3 Wave Speed vs Upper Level Zonal Wind, Northern Upper level zonal wind (m/s) y =.11x R² = 1E- Fig., Wave Speed vs. Upper Level Zonal Wind, Northern 3 Wave Speed vs Upper Level Zonal Wind, Southern y =.126x Upper level zonal wind (m/s) R² =.1813 Fig. 6, Wave Speed vs. Upper Level Zonal Wind, Southern

10 Wave Speed Wave Speed Wave Number vs (Wave Speed - U) NH Wave Speed y =.91x R² =.12 Fig. 7, Wave Number vs. (Wave Speed U), Northern Wave # vs (Wave Speed - U) SH Wave Number y = -.48x R² =.2 Fig. 8, Wave Number vs. (Wave Speed U), Southern

11 Wave Number Wave Number Wave Number vs., Northern R² = Sep 19-Sep 29-Sep 9-Oct 19-Oct 29-Oct 8-Nov 18-Nov 28-Nov Fig. 9, Wave Number vs., Northern 7 6 Wave number vs., Southern R² = Sep 19-Sep 29-Sep 9-Oct 19-Oct 29-Oct 8-Nov 18-Nov 28-Nov Fig. 1, Wave Number vs., Southern

12 Amplitude Amplitude 3 Amplitude vs. Wave Number, Northern Wave Number Fig. 11, Amplitude vs. Wave Number, Southern Amplitude vs. Wave Number, Southern Wave Number Fig. 12, Amplitude vs. Wave Number, Northern

13 Amplitude (m) Amplitude (m) Amplitude vs., Northern R² = Sep 19-Sep 29-Sep 9-Oct 19-Oct 29-Oct 8-Nov 18-Nov 28-Nov Fig. 13, Amplitude vs., Northern Amplitude vs., Southern R² = Sep 19-Sep 29-Sep 9-Oct 19-Oct 29-Oct 8-Nov 18-Nov 28-Nov Fig. 14, Amplitude vs., Southern

14 Maximum -hpa wind speed (m/s) Maximum -hpa Wind Speed vs., Northern R² = Sep 19-Sep 29-Sep 9-Oct 19-Oct 29-Oct 8-Nov 18-Nov 28-Nov Fig. 1, Maximum hpa Wind Speed vs., Northern 6 Maximum -hpa Wind Speed vs., Southern R² = Sep 19-Sep 29-Sep 9-Oct 19-Oct 29-Oct 8-Nov 18-Nov 28-Nov Fig. 16, Maximum hpa Wind Speed vs., Southern

15 Amplitude/Zonal Wind 1-Sep 22-Sep 29-Sep 6-Oct 13-Oct 2-Oct 27-Oct 3-Nov 1-Nov Amplitude (m) Zonal Wind (m/s) Amplitude vs Zonal Wind, Northern Amplitud e U Fig 17, Amplitude vs. Zonal Wind, Northern 3 Amplitude vs Zonal Wind, Southern Amplit ude U Uuppe r Fig. 18, Amplitude vs. Zonal Wind, Southern