Global Seasonal Phase Lag between Solar Heating and Surface Temperature

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1 Global Seasonal Phase Lag between Solar Heating and Surface Temperature Summer REU Program Professor Tom Witten By Abstract There is a seasonal phase lag between solar heating from the sun and the surface temperature across the globe. The time of year at which the amount of solar heating is maximal and minimal lag the maximal and minimal surface temperatures by several weeks. We have determined this phase lag precisely using historical data from 74 data stations compiled from the NOAA Database. Our measured phase lag of days is observed to be similar for many locations around the globe. Here we explore a possible explanation for this phase lag through a thermal conduction model. This model however gives much larger than the measured values. We show the failure of the thermal conduction model and that there is a significant correlation of phase lag with latitude. For certain latitudes there appears to be a minimum phase of about 0.06 a yearly cycle. 1 Introduction Seasons As the Earth travels in its orbit around the sun the relative angle of the sun at different latitudes with respect to the surface changes. This is what gives rise to the seasons we experience. As this relative angle goes to a minimum or maximum the amount of solar heating becomes minimal and maximal as well. These times of the year are known as solstices and the times of the year at which these angles are equal are known as equinoxes. For mid-range latitudes in the northern and southern hemispheres solar heating goes through one cycle per year. This means that there is one time per year with a maximum or minimum angle. We know these dates in the northern hemisphere as the summer and winter solstices, June 21 st and December 22 nd. For latitudes closer to the equator solar heating goes through two cycles per year. The angle of the sun relative to latitudes in the equator range is minimal at the spring and fall Equinox and maximal at the summer and winter solstice. Based on these observations, one can infer, in addition to solar heating having dominating first and second harmonic behavior, the temperature will have dominating first and second harmonic behavior as well. A glass of liquid As a starting point we would like to discuss a simple example involving heating of a glass of liquid via an oscillating heat source. The heat source is turning on and off at some constant driving frequency ω. For small w the temperature of the cup and incoming heat radiation are in phase and for large ω the temperature and heating are exactly π/2 out of phase. For the intermediate range of driving frequencies the phase is between 0 and π/2. Thus, there are a range of possible phase lags that one can observe the cup of liquid to have based solely on the driving frequency of the heat source. Additionally this system has a characteristic frequency ω 0 that governs the systems intrinsic response to driving. This characteristic frequency is equal to 1/

2 (Relaxation time). Thus the characteristic frequency is a feature dependent upon the property of the liquid or material being heated. To conclude this simple example, it is known that, in general, any system that has a relaxation time displays a range of phase lags dependent upon the driving frequency of the system and has a preferred frequency at which the intrinsic resonant response is maximal. In addition the phase lags are appreciable only if the driving frequency is within an order of magnitude of the relaxation rate. [1] If we were to use this model to explain the earth s phase lag between temperature and heating we would expect the relaxation time of the earth to match the period of its orbit or driving frequency of one year. If the earth could be isolated from its orbit around the sun and all heat radiation, it would only take a year to cool off. This seems to be an implausible conclusion. The typical explanation of phase lags from linear response theory fails and we are thus led to look for a broader explanation of this phenomenon, which brings us to the thermal conduction model. Thermal Conduction Model The thermal conduction model treats the surface of the earth as a thick and infinite uniform conducting slab of material with thermal diffusivity ζ. [1] In this case we will assume the slab to be z>0 and the periodic heat to be coming from z. When we inject the slab with periodically varying heat we obtain a temperature profile T(z,t) of the form: The heat diffuses into material obeying the diffusion equation. Assuming the above temperature profile the diffusion equation can be solved for the phase lag in the following way; The diffusion equation reads, and can be rewritten as when the assumed arbitrary sinusoidal temperature profile is used. This yields the following result for q, In order for the response to decay as one penetrates further into the medium the positive square root must be used. Finally at the surface we know the heat flux is proportional to the temperature gradient. We obtain the following by substituting exponential forms for -1, i and q. The heat flux and temperature phase vectors, which are rotating clockwise in time, have a phase lag between them of π/4 radians, 1/8 or 12.5% of a year. The result is independent of the driving frequency and the thermal diffusivity of the material and does not require the same tuning as the glass of liquid model, thus providing a broad and nice explanation of this phenomenon.

3 2 The Data Figure 2: A map of weather data points from the National Climatic Data Center Online. Figure 1: Map of available Weather Stations To analyze the yearly phase lag one needs to obtain daily temperature records over a period of many years. We chose 74 stations with approximately 50 or more years of data. Data was acquired from the National Climatic Data Center [2]. The above map in Figure 1 shows all of the available weather stations across the globe. The data we used in our analysis was the Daily Minimum Temperature; this is the lowest measured temperature at a given location on a given day. The weather stations typically record the temperature every hour, so this statistic is the lowest of those over a 24 hour period. The locations we chose were typically away from major cities to avoid man made affects on the temperatures. Most of the stations we chose to use were US weather stations. This is simply because the weather data from other places around the globe was not sufficiently long enough to provide reliable analysis. We tried to choose weather stations with at least 60 years of data from a variety of different geographies. One of the more important parts of this project was developing an algorithm using Mathematica that could take the many years of data and analyze the daily average. Once this algorithm was complete more important data could be developed including computation of phase lags and detailed error analysis.

4 3 Results and Discussion Summer REU Program Average Temperature Data Figure 3: A plot of Daily Average Low Temperature vs. Day of Year The graphs in Figure 2 show the average temperature data for all of the locations we chose to analyze. Each color represents a different station. As expected the graphs all have minimum and maximum values at about the same time of year; and are fairly independent of the geography of the location. It is important to note that some of the data does appear to be affected by location, an early sign that phase lags might have some dependence on geography. The graphs are scaled on the x-axis from 0-1, 0 being the beginning of the year and 1 being the last day of the year. The locations towards the top of the graph that do not fluctuate much at all are locations that are closer to the equator. Notice the obvious presence of a second harmonic in some of the data. The average temperatures for the locations closer to the equator reach a maximum and minimum twice per year whereas the mid-range latitudes go through only one cycle of maximum and minimum temperatures. Also notice the one location that appears flipped relative to the others; this is a location in Antarctica. As we expect summer and winter in the Southern Hemisphere occur at times opposite that of the northern hemisphere. The next part of the data analysis was acquiring fitted models for all of the average data so we could calculate the phase lags.

5 Fitted Models Figure 4: All Average Temps graphs superimposed with fitted models. In order to obtain proper fits for the average data we used a cosine model with two harmonics. As we can see in Figure 3 the fits strongly agree with the above datasets. Notice even for the data sets that have slightly strange peaks, clear signs of geographic effects, that the models still fit the data very well. The fits were found using the following form: This can be rewritten as: From these two equations the phase lag φ for the first harmonic term could be found by using the following equation:

6 Phase Lags Figure 5: Phase Lags with Error Bars For several locations we were able to reproduce the results for phase lags that were obtained by a previous REU student a few summers ago. The phase lags were 4-6% less than what the thermal conduction model predicts it would be. The phase lags were calculated to be nearly the same for all the locations across a variety of geographic variables but all still considerably less than the value of 12.5% of a yearly cycle that the thermal conduction model predicts. The graph in Figure 4 shows the results for the phase lags for each of the 74 Weather Stations we chose to analyze. As we can see the error bars are generally very small compared to the phase lags. One of the important things we noticed here was that there is a station to station discrepancy between the phase lag values. The error bars for some of the data points however are clearly out of the range of the others which supports our notion that there are geographic variables affecting the phase lag values. Some of the geographic variables which might be suspect in altering these values include some of the locations being islands, latitudes of the locations and elevations of the landlocked locations. Latitude seems to be the most important factor for the phase lags as we will see in the next section.

7 Phase Lags vs. Latitude Figure 6: Values of Phase Lag vs. Latitude of Location The graph in Figure 5 shows the correlation between the latitude of each location and the phase lag value at that location. As we can see there seems to be a relationship between these two values with the exception of a few locations. As the latitude decreases in the Northern Hemisphere and gets closer towards the equator the phase lag decreases. Unfortunately we a lack of solid set of data for weather stations close to the equator and in the Southern Hemisphere. In order to confirm this relationship we need to obtain more data for these regions. Data for these regions proved to be hard, if not impossible to find because we were trying to use at least 50 years of data. Additionally the fluctuations in average temperature for locations close to equator have amplitudes of only 4 degrees in some cases. Temperature data is generally recorded to the nearest degree so for these low latitude locations this could severely affect the phase lag calculations. We are confident that there is strong correlation between latitude and phase lag, and that there is an explanation for this correlation in terms of some simple physics.

8 4 Conclusions In this study of phase lags between annual solar heating and the temperature cycle we have discovered a few new results and shown that a thermal conduction model is an insufficient theory for this phenomenon. We observe a very well defined response to the annual heating cycle for a given location over the period of a year. Additionally this response involves a phase lag that is systematically too small to be explained by the thermal diffusion model, thus showing the thermal conduction model is insufficient. The discrepancies between some of the values obtained are far outside of the error bars and shows there is dependence on geographic variables, of which the most dominating appears to be latitude of the location. This is a new result as it was previously thought that the phase lag values would be within the same range independent of locations. From what we observed this is not the case and provides a new interesting feature to this problem that was not previously known. We believe that from this correlation of phase and latitude we can find a simple theory in terms of geometry arguments or some other fundamental principles that sufficiently explains these phase lags. Resources: 1.) Tom Witten, University of Chicago, James Franck Institute. Paper on Phase Lag Problem, On the meaning of the Seasonal Phase Lag. 2.) National Climatic Data Center, Online Database: