Using the frequency of Teletherm flickering to map the propensity for climatic regimetransitions

Benjamin F. Emery,1, ∗ Christopher M. Danforth,1, † and Peter S. Dodds1, ‡

1Vermont Complex Systems Center, Computational Story Lab,The Vermont Advanced Computing Core, Department of Mathematics & Statistics,

The University of Vermont, Burlington, VT 05401.(Dated: November 23, 2018)

The shifting global climate has long been known as a complex system consisting of a vast array ofinteracting dynamical components. The thorough studies on many of the changing characteristicsof the climate have been insightful, but much of the picture remains unpainted. A 2015 studyintroduced the Summer and Winter Teletherms, the dates of the on-average highest and lowesttemperature. These fundamental markers were measured on a range of timescales, with 25 yearsbeing the length of time found to be the most informative when it came to expressing changes in theTeletherms over time. Observation of the progression of these Teletherms over time showed that theTeletherms tended to jump discontinuously between regimes, and in some locations this jumpingwas more prominent than others. In this study, we investigate the variation in the prominence of thisjumping between locations by introducing two measurements, the Summer and Winter FlickeringCoefficient, which express the degree of this behavior exhibited in a specified location for both theSummer and Winter Teletherms. We find that on average, locations have a higher Winter FlickeringCoefficient than Summer. Within the contiguous United States, there is a positive between thisdifference and Longitude, and on a global scale, we find that higher latitudes also correlate withgreater differences between the two values. A number of other more complex geographic patternsare observed, as well as strong regional continuity.

I. INTRODUCTION

Increasingly dramatic changes in the global climatehave resulted in droughts, food and water crises, andmany other disasters. These have often occurred onglobal scales, but the severity of their effects on localcommunities varies geographically. Locally predictingthese events and their severity also varies in difficulty.Developments in mathematical meteorology have shownthat atmospheric convection can be modeled as thechaotic attractor of a dynamical system, with the eigen-values of the Jacobian matrix correlating to the difficultyof weather prediction at a given set of coordinates.

A previous study at the University of Vermontby Dodds et al introduced the Summer and WinterTeletherms as statistical measures of the hottest andcoldest days of the year [1]. These Teletherms werecomputed over time intervals, most commonly of 25years in length. During the analysis of these Telethermsover many intervals centered at consecutive years, themajority of locations displayed discontinuous jumpingof the date of either their Winter Teletherm, SummerTeletherm, or both. When they occurred, these jumpswere between two or more narrow windows of dates. Theamount of this jumping varied widely between differentlocations.

In the present study, we propose a method of quanti-fying the amount of such discontinuous jumping in these

∗ bfemery@uvm.edu† cdanfort@uvm.edu‡ pdodds@uvm.edu

Teletherm dates at each location, and investigate thepattern of their spatial distribution. We call these mea-surements Flickering Coefficients, after use of the termflickering by Scheffer et al to describe the movement ofa dynamical system back and forth between separatebasins of attraction [2]. Scheffer discusses this flickeringphenomenon in dynamical systems as a warning sign ofa possibly imminent critical transition to a new basin.Dodds et al indeed observe this flickering behaviorbetween two dates usually lead to a long-term change inthe Teletherm for that location.

Like Teletherms, there are separate Winter andSummer Flickering Coefficients. Both of these measure-ments, as well as the pair-wise differences between them,show quasi-continuous spatial changing, with someoutliers. As we will later discuss, there exists a powerfulinterpretation of these Flickering Coefficients as theyrelate to the prediction of extreme weather events.

II. DATA

We consider measurements of daily temperatureextremes from 3038 observatories worldwide. 1113 ofthese datasets are located within the contiguous UnitedStates, drawn from the United States Historical Clima-tology Network (USHCN) data set (version 2.5 through2012). The remainder are drawn from the Global His-torical Climatology Network (GHCN) data set (version3.22 through 2016) [3]. The positions of these stationsare shown in Figure 1 for the US dataset and both setscombined.

Typeset by REVTEX

2

FIG. 1. Locations of 3038 observatories represented by theUSHCN and GHCN datasets. Coverage varies widely, fromvery dense coverage in North America to nearly empty cover-age in Africa and South America.

We index the dates of a year from 1 to 365, adjust-ing by one day on leap years to maintain a 365-day year.We initialize day 1 of the year at January 1 for analysesof maximum temperature, and at July 1 for analyses ofminimum temperature.

The years covered by these measurements vary widelybetween observatories. Some begin measurement as earlyas 1850, most continuing to measure through present day,but some phasing out. Notably many datasets from Aus-tralian observatories phase out sometime in the 1990s.For our analyses, we require that the datasets cover atleast 65 consecutive years.

III. METHODS

A. Teletherm dates

At each location, we compute the 25-year summer andWinter Teletherms. We compute these for every intervalof 25 years for which the station has coverage.

To compute a 25-year Teletherm, we find the averagehigh and low temperature for each day over the 25 years.To include leap years in this average, we consider the tem-peratures on February 28 to be the average of those onFebruary 28 and 29, and remove February 29 from thatyear. We then smooth these averages using a one dimen-sional Gaussian kernel with σ = 3.5 days, correspondingto a kernel width of 15 days. We conduct this convolu-tion using the Astropy commands Gaussian1DKerneland convolve[4]. The smoothed and unsmoothed aver-ages are shown for an example location and time intervalin Figure 2, and the Teletherm date is indicated with avertical bar. The date of the highest smoothed max-imum temperature is the Summer Teletherm for thatinterval, and the date of the lowest smoothed minimumtemperature is the Winter Teletherm. The smoothingprocess allows us to defend our treatment of the extraday on leap years as not significantly impacting the com-puted Teletherm. We identify the interval over which a

FIG. 2. Daily high temperatures averaged over 25 years (blackdots), along with a smoothed curve generated by a Gaus-sian kernel (solid red) in Yellowknife, Northwest Territories,Canada. The Summer Teletherm date is shown with a ver-tical black line to the curve, indicating the date where thesmoothed curve reaches its highest point for the specified 25-year time interval.

Teletherm is calculated by the final year in the interval.

B. Flickering Coefficients

The timeseries of these Teletherm dates, for bothsummer and winter, often display discontinuous jumpingbetween prominent modes of the annual temperatureprofile. In some instances, the switch would take placeonce, and the Teletherm would remain in the new regimefor an extended period of time. In other instances, theTeletherm date would swap between regimes multipletimes. The latter behaviour is reminiscent of the “flick-ering” discussed by Scheffer et al as a signal indicatingan upcoming critical transition. Timeseries examples ofthis behavior are shown in Figure 3.

We quantify this dynamical behavior with two mea-surements, which we call the Summer and WinterFlickering Coefficients. The Flickering Coefficients,whose calculation is shown below, is effectively the pro-portion of years where the Teletherm date jumps by morethan seven days from the previous year’s Teletherm date.

FC = # jumps > 7 days# years measured

We choose to analyze this measurement, rather than con-ventional measurements of variation such as standarddeviation to emphasize the presence of discontinuousshifts and neglect smooth changes over time. Addition-ally, we seek to measure the frequency of the flickeringrather than the distance of the jumps.

3

FIG. 3. Time series plots for the winter and summer 25-year Teletherm dates (dark gray squares) in Yellowknife, NWT, Canada.This location, along with many others, displays discontinuous jumps between Teletherm regimes. The Flickering Coefficient isdefined by the number of times that such a shift took place per year that a Teletherm can be calculated.

IV. ANALYSIS AND RESULTS

A. Flickering Coefficient behavior

We seek to investigate the properties of the distribu-tion of Flickering Coefficients as measured at differentlocations worldwide. In Figures 4A and 4B, we map theSummer and Winter Flickering Coefficients. The pointson each map are colored according to the coefficient alongthe same gradient, shown on the bar alongside the maps.Additionally, we provide histograms of the collection ofFlickering Coefficients, also colored with the same schemeas the points on the maps. Geographical coverage is par-ticularly high for our data set in the contiguous UnitedStates, but is also sufficiently high in Europe, areas ofAsia, and the Australian coast. Figure 5 provides a closerview of the contiguous US, where coverage is the highest.

In areas where coverage is high, we see that geograph-ical variation in Flickering Coefficient is, for the mostpart, locally smooth, sometimes with noticeable transi-tion boundaries between regions. This behavior hints atthe ability to predict the Flickering Coefficient in loca-tions where measurements have not been recorded forlong enough for a coefficient to be calculated, if the loca-tion is in the proximity of measured values in multipledirections.

As can be seen in the histograms accompanyingthe maps, Summer Flickering Coefficients (SFC’s) areon average lower than Winter Flickering Coefficients(WFC’s). Distributions of both measurements have asubstantial right skew, but distributions of the SFC’sshow much near-zero clustering, where as the distribu-tions of WFC’s has a peak around 0.6 for the global dataand 0.4 for the United States data. In the following sub-section, we consider the difference between each WFCand its corresponding SFC.

The maps reveal some notable geographic patterns in

Flickering Coefficients. In Europe, we find a SFC’s clus-tering around 0 in Germany and Poland, yet those sameareas have WFC’s around 0.15. We see similar behaviorin Russia along the 60N Latitudinal axis, especially westof 60W. Japan’s SFC’s similarly cluster around 0, but itsWFC’s remain lower than 0.10.

In the United States, SFC’s are high in the southeastcorner, between 0.10 and up to 0.30. Low SFC’s becomemore frequent further north, with two major clusters of 0and near-0 SFC’s. One such cluster is centered on Idahoand includes bits of Montana, Oregon, and Washington.The other includes Minnesota, Wisconsin, and pieces oftheir surrounding states. We additionally observe thatlocations very close to the western coastline have high-er SFC’s than locations slightly further east. WFC’s arehigher overall with similar geographic patterns. The highWFC’s in the southeast corner extend northbound as aband stretching from Alabama, through Tennessee andreaching West Virginia along the Appalachian mountainrange.

B. Differences between winter and SummerFlickering Coefficients

At each location, we subtract the Summer Flicker-ing Coefficient from its winter counterpart. The globalresults are mapped in Figure 6, and the results for thecontiguous United States are mapped in Figure 7. As thehistograms provided with each map show, the distribu-tion of this quantity appears to be Gaussian.

The global collection of measurements of this differ-ence has a mean of 0.02243, with a standard deviationof 0.09035. A one-sample T-test for the significanceof the difference between this mean and zero returnsp = 2.13 ∗ 10−41, revealing a very significant departurefrom 0. Given that our data lacks coverage of nearly allof Africa, South America, and many other regions, we

4

FIG. 4. A: Summer Flickering Coefficients and B: Winter Flickering Coefficients plotted in all locations found in the GHCNand USHCN for which enough coverage existed for the necessary calculation. The value of the coefficient expressed by the colorof each point can be determined with the shared colorbar to the right of the maps.

can’t extend this result globally. If we, however, restrictour conclusions to the regions where we have coverage,we can reasonably expect the mean of this quantity ofall locations to be greater than zero. The distributionexclusive to the contiguous US has a mean of 0.02239, astandard deviation of 0.06180, and the same test returnsp = 1.12 ∗ 10−31, so the same conclusion holds for thecontiguous United States in particular.

The maps of the Flickering Coefficients and their dif-ferences seem to display a slight favoring toward higherWFC-SFC values at higher latitudes. To investigate this

further, we plot this value against latitude in Figure 8.Because of the scant southern hemisphere coverage, weonly include the northern hemisphere in this analysis.A least-squares linear regression test produces a line ofslope 0.00089 with p = 1.083 ∗ 10−9, with r = 0.114.We hypothesize that, among other causes, changes inthe snowfall pattern between years can cause dramaticshifts in the Teletherm date. Snow cover can dramatical-ly decrease the atmospheric temperature by increasingthe land’s albedo. The date of the first substantial snow-fall can thus have a substantial impact on the arrival of

5

FIG. 5. A: Summer Flickering Coefficients and B: Winter Flickering Coefficients from only the USCHN dataset.

the coldest weather that year. Areas further from theequator, those that normally have sustained snow cover,are thus likely to have higher WFC’s.

In addition to being positively correlated to latitude,the distribution of these differences displays complex geo-graphical patterns similar to the distributions of SFC’sand WFC’s alone. Germany, Poland, and Russia are alldominated by positive differences, as we would expectfrom our discussion of them in the previous subsection.Southern sections of Europe, Asia, and Australia aredominated by points of nearly 0 difference, as well asmost of the US. From the US map we can see thatthere is a notable band of positive differences along theAppalachians. Apart from this, from west to east, nega-tive differences become less frequent. This, as well as theband of high values near the east coast are reflected in

a linear regression against longitude, shown in Figure 9.This least-squares regression returns a slope of 0.00076,with r = 0.178 and p = 2.288 ∗ 10−9.

V. CONCLUDING REMARKS

Inspiration for this study spawned from the provoca-tive results in the 2015 paper on Teletherms by Doddset al. The discontinuous movement of the 25-yearTeletherms over time at such a notable proportion oflocations was yet another demonstration of the earth’sclimatic behavior as a complex dynamical system. Thisbehavior indicated a potentially useful measurementof Teletherm dynamics that varies geographically. Wehoped mapping such a measurement would provide a

6

FIG. 6. The difference between winter and Summer Flickering Coefficients at each location. The color of each point expressesthis difference, with the correspondence indicated by the colorbar. The histogram bars follow the same color scheme as themap.

greater understanding of the changing climate, namelyan improvement in foresight about its effects on theoccurrence of locally specific extreme events. The resultsshow us that certain regions are much more susceptibleto regime swapping than others. If we follow the logicof Scheffer et al, this may lead us to believe theselocations are likely transition to a new regime for apotentially long stretch of years. This, combined withother knowledge about Teletherm dynamics, can helpus predict future dates of extreme heat, or dangerouscold-weather events. Regions with low FC’s are likely toobserve these events around the same time each year,where as those with high FC’s can expect a perhapssudden shift to a new consistent time of occurrence.

Our analyses open the door for many more directionsfor further studies involving the measurement and inves-tigation of the dynamical behavior of the Teletherms.Supplementary to our analyses done with FC differencesand coordinates, one could investigate their relationshipswith other likely contributers, such as elevation anddistance to a coastline.

In addition to investigating the relationship betweenFC’s and various geographical characteristics, there areother related measurements that can describe featuresof the dynamics of the Teletherms. One such possibility,and a likely next step for our team, would be to dive intothe behavior of the Teletherm Periods. The TelethermPeriod for a particular location and interval of years,as defined by Dodds et al, is the set of dates for whichthe smoothed temperature comes within 2% of theTeletherm temperature, as it is measured with respectto the range of the smoothed temperature curve.

Often accompanying the discontinuous jumps in theTeletherm dates are bifurcations in the Teletherm period,where it splits and becomes temporarily non-contiguous,corresponding to a set of years where smoothed temper-ature profile has multiple modes competing to containthe Teletherm date. This phenomenon is present in theTeletherm timeseries shown in Figure 3. Measurementsusing these bifurcations might include the average size ofinterior gaps in the Teletherm periods, where contiguousTeletherm periods have an interior gap size of 0 days.The average number of interior gaps per year would beinformative as well.

Changes in the Flickering coefficent over time arealso of great interest, as such statistics would hold newinformation about how the behavior of Teletherms haschanged alongside other notable descriptors of globalclimate, namely temperature, ocean acidification, andglacial mass. Regrettably, availability of data to usefor this is dubious, since so few observatories have beencollecting adequate data for long enough.

Additionally, many insights relating to the originalTeletherm study remain undiscovered. The methods ofthis study uncovered the Teletherms in roughly 5, 000locations not originally considered. As the conclusionsof that study suggest, this data may have provided newopportunities to formulate and test models relating toglobal climate analysis.

7

FIG. 7. The difference between winter and Summer Flickering Coefficients at each location in the USCHN dataset.

FIG. 8. The difference between the Winter and Summer Flick-ering Coefficients regressed against the latitude of the locationwhere they are measured. Due to scarce and geographicallynon-diverse data in the southern hemisphere, only stationsin the northern hemisphere are included for this regression.The linear trend line has a slope of 0.00076, and the least-squares regression test yields r = 0.114 and p = 2.288 ∗ 10−9.The 95% confidence prediction band is indicated by the twodashed lines above and below the fit line.

ACKNOWLEDGMENTS

The authors are grateful for support and helpful feed-back from the members of the Computational Story Lab.

FIG. 9. The difference between the Winter and SummerFCs in the contiguous US regressed against the longitude ofthe location where they are measured. The linear trend linehas a slope of 0.00089, and the least-squares regression testyields r = 0.178 and p = 1.083 ∗ 10−9. The dashed lines onceagain indicate the 95% confidence prediction band.

8

[1] P. S. Dodds, L. Mitchell, A. J. Reagan, and C. M. Dan-forth, PloS one 11, e0154184 (2016).

[2] M. Scheffer, J. Bascompte, W. A. Brock, V. Brovkin, S. R.Carpenter, V. Dakos, H. Held, E. H. Van Nes, M. Rietkerk,and G. Sugihara, Nature 461, 53 (2009).

[3] I. D. B. K. S. M. K. T. X. Y. S. A. R. R. R. V. B. Menne,M.J. and T. Houston, “Global historical climatology net-work - daily (ghcn-daily), version 3.22,” (2012).

[4] T. P. Robitaille, E. J. Tollerud, P. Greenfield, M. Droett-boom, E. Bray, T. Aldcroft, M. Davis, A. Ginsburg, A. M.Price-Whelan, W. E. Kerzendorf, et al., Astronomy &Astrophysics 558, A33 (2013).