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RISE UNDERGRADUATES FIND THAT REGIMECHANGES IN LORENZ’S MODEL ARE PREDICTABLEBY ERIN EVANS, NADIA BHATTI, JACKI KINNEY, LISA PANN, MALAQUIASPEÑA, SHU-CHIH YANG, EUGENIA KALNAY, AND JAMES HANSEN

Note from the BAMS editors: This article was origi-nally reviewed and submitted to BAMS before the stu-dent section editorial board was formed, but is includedhere as an outstanding example of educational oppor-tunities now available to undergraduates.

he summer of 2002 marked the beginning of theResearch Internships in Science and Engineering(RISE) program. Funded by the National Science

Foundation (NSF), the A. James Clark School of En-gineering, and the University of Maryland, and co-ordinated by the Women in Engineering Program,RISE worked to build an extensive network of womenfaculty, science and engineering research profession-als, graduate students, and undergraduates at all lev-

els. The program built this network through an8-week summer research experience for “rising” jun-ior and senior undergraduates. The goal was to en-courage all participants to remain in the fields of sci-ence and engineering and to pursue graduate degreesin these fields.

By engaging 20 undergraduate junior and seniorRISE scholars in team-based research projects coor-dinated by female faculty, the program introducedfemale students to women mentors and role modelswhile providing high-quality opportunities to en-hance their research knowledge and skills. RISEscholars were equipped with advanced training inteam skills, interpersonal communication, andproject management. They were also able to becomea part of the hierarchy of female mentorship by in-teracting with a group of incoming freshmen stu-dents. By sharing their experience as students in sci-ence and engineering and as RISE scholars, theybecame role models to the younger students. Thesummer’s concluding event was the RISE ResearchSymposium, where research teams gave oral presen-tations of the results of their research, as well as aposter exhibit documenting their research activitiesand results. Coordinating faculty and representativesfrom the A. James Clark School of Engineering andthe College of Mathematics and Physical Sciences, aswell as staff from the NSF, were among those attend-

AFFILIATIONS: EVANS, BHATTI, AND PANN—RISE scholars at theUniversity of Maryland at College Park, Summer 2002; KINNEY—RISEscholar at the University of Maryland at College Park, Summer 2002,and Texas A&M University, College Station, Texas; PEÑA AND YANG—RISE graduate advisors; KALNAY—RISE faculty mentor, and Universityof Maryland at College Park; HANSEN—Massachusetts Institute ofTechnology, Cambridge, MassachusettsCORRESPONDING AUTHOR: Eugenia Kalnay, Dept. of Meteorol-ogy, University of Maryland at College Park, 3431 Computer andSpace Sciences Bldg., College Park, MD 20742-2425E-mail: ekalnay@atmos.umd.edu

©2004 American Meteorological Society

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ing the symposium. R. Colwell, director of the NSF,provided the keynote address.

One of the teams worked on atmospheric predict-ability. The purpose of this note is to describe the ex-perience and results obtained by this team in orderto encourage similar programs to attract women andminorities into graduate studies in the geosciences.One of the RISE scholars described her experience asfollows: “As an intern in the RISE program, my mainexpectation was to gain familiarity with the researchprocess. Without prior research experience, I wasunsure if graduate school was a realistic option for me.The RISE program allowed me to make my final de-cision to pursue a graduate degree and gave me con-fidence in my ability to contribute to a researchproject. I was assigned to a project in the field of me-teorology, a field about which I had little or no knowl-edge. Through the guidance of our faculty mentor,Dr. Eugenia Kalnay, and graduate advisors MalaquíasPeña and Shu-Chih Yang, my team and I were ableto first understand problems involved with weatherprediction, and then apply our new knowledge toresearching a method of weather prediction. Inter-preting results, accepting that actual results may notagree with expected ones, and exploring new pathsthat the results lead to, are all exciting componentsof the creative process of research that my team and Ihad the opportunity to engage in. The fact that wewere able to contribute to the discovery of new results

helped me to decide to continue participating in re-search in graduate school.”

PREDICTABILITY STUDY OF THE LORENZ(1963) MODEL. Although the four RISE internswere selected because of their outstanding mathemati-cal, physical, and computer science skills, three ofthem had no background in meteorology, and the factthat the research internship had to be completed in8 weeks imposed a significant challenge. The team wasgiven a problem: become familiar with the famousLorenz (1963) model, and explore its predictabilityusing breeding (Toth and Kalnay 1997; Kalnay 2003),an algorithm chosen for this project because of its sim-plicity. The Lorenz model equations are

(1)

,

where the parameters s = 10, b = 8/3, and r = 28, cho-sen by Lorenz, result in chaotic solutions (Fig. 1).This model has been very widely used as a prototypeof chaotic behavior and an example of lack of long-term predictability (e.g., Sparrow 1982; Tsonis 1992;Kalnay et al. 2002). The stability properties and thedependence of the forecast error growth on the ini-tial conditions have been previously studied (e.g.,Nicolis et al 1983; Nese 1989; Elsner and Tsonis 1992;Palmer 1993), but we are not aware of studies aboutthe prediction of the occurrence of regime changesand their duration. The students were given as a tem-plate a Matrix Laboratory (MATLAB) program of acoupled fast–slow Lorenz model written by J.Hansen, from which they unraveled the classicLorenz model code and learned how to run and plotits results. They were asked: “Imagine that you are aforecaster living in the Lorenz attractor. Everybodyin the attractor knows that there are two weatherregimes, which we could denote as ‘Warm’ and ‘Cold’(see Fig. 1), but the public needs to know whenchanges in regime will happen and how long will theylast. Can you develop simple forecasting rules to alertpeople about imminent changes of regime?”

The students implemented breeding, a methodused to estimate forecast errors in weather models.Bred vectors are simply the difference between two

FIG. 1. Solution of the Lorenz model equations (1) over1500 time steps, showing a “warm” regime with posi-tive values of x and y, and a “cold” regime with nega-tive values of x and y. The solution typically remains forseveral loops in each regime before changing to theother regime.

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model runs, the second originating from slightly per-turbed initial conditions, periodically rescaled (Fig. 2).The amplification of the bred vectors can be used toidentify regions of high error growth within theattractor. The Lorenz model used in this project wasintegrated using a fourth-order Runge–Kutta timescheme with a time step of Dt = 0.01. The bred vectorswere obtained from a second run with the same modelstarted from an initial perturbation δx0 = (δx0, δ y0, δ z0)added to the control at time t0. Every eight time stepsthe vector difference δ x between the perturbed andthe control run was rescaled to the initial amplitude

and added to the control run (Fig. 2). The bred vec-tor amplification factor was defined as the size of thebred vector after n = 8 steps divided by its original size|dx|/|dx0|, and the growth rate as

.

The students plotted the observed bred-vector growthon the Lorenz attractor in order to explore its predict-ability (Fig. 3). Red indicates that during the last eightsteps the perturbation growth rate g was larger than0.064 (i.e., the size of the bred vectors grew by a fac-tor of 1.67 or more in eight time steps), whereas blueindicates a negative growth rate, meaning that the per-turbations are actually decaying. The results shownin this figure were very promising because they sug-gested that bred vector growth would allow estimat-ing regions of high and low predictability of theattractor.

The students then examined the bred-vectorgrowth for patterns of predictability. They found thatplotting the growth rates on the evolution of the vari-able x(t) provides a means to predict when the modelwill enter a new regime, and also how long the newregime will last. Figure 4 illustrates the “forecastingrules” that the students developed by inspection.

• Rule 1: When the growth rate exceeds 0.064 overa period of eight steps, as indicated by the presenceof one (or more) red stars, the current regime willend after it completes the current orbit.

• Rule 2: The length of the new regime is propor-tional to the number of red stars. For example, thepresence of five or more stars in the old regime,indicating sustained strong growth, implies thatthe new regime will last four orbits or more (seeFig. 5 for the relationship between number of redstars and the duration of the new regime).

After the RISE internship had been completed, andthe results presented at the RISE Research Sympo-sium, Evans (supported by the School of Engineering)and Peña carried out an objective verification of thesesimple forecasting rules. Table 1 is the contingencytable for the categorical Rule 1 that forecasts the oc-currence of a regime change during the followingorbit. Table 2 is the corresponding contingency table

FIG. 2. Schematic of the construction of bred vectors,which are the difference between a perturbed and a“control” (unperturbed) solution. Every few (in thiscase, eight) steps, the difference, rescaled to the origi-nal size and added to the control forecast, becomes theinitial condition for the perturbed forecast. The ratiobetween the initial and the final size is the amplifica-tion of the bred vector during that interval.

FIG. 3. The Lorenz “butterfly” attractor with bred-vec-tor growth over eight steps. The table indicates therange of the growth rate corresponding to each color.

523APRIL 2004AMERICAN METEOROLOGICAL SOCIETY |

In summary, the RISE students succeeded in pro-viding the mythical inhabitants of the chaotic Lorenzattractor with robust prediction rules that would al-low them to be prepared for changes in regime andindicate how long the new regime would probablylast. While in this process, the undergraduate womenlearned that they were able to both perform and en-joy research, strengthening their motivation to pur-sue research careers in science and engineering.

ACKNOWLEDGEMENTS. We are very grateful toLinda Schmidt, Janet Schmidt, and Paige Smith for conceiv-ing and obtaining funds for the RISE program, for the sup-port of the National Science Foundation and the University

FIG. 4. Time series of the variable x versus time step, withbreeding cycles of eight time steps, with colored stars indicat-ing the bred-vector growth, as in Fig. 3. Each panel shows 4000steps, corresponding to 500 breeding cycles.

FIG. 5. Observed number of cycles in the new re-gime for a given number of red stars in the oldregime. The numbers indicate the number ofpairs observed, with blanks indicating no ob-served pairs. Dashed lines indicate the specificrule verified in Table 2, although other combi-nations yield similarly high verification scores.

for Rule 2, where the presence of four stars or less in-dicates that the new regime will only last up to threeorbits. Figure 5 shows that there is a strong relation-ship between the number of red stars in the old re-gime and the duration of the new regime. The verifi-cation scores obtained for the rules, with hit rates over90%, threat scores over 80%, and false-alarm rates ofless than 10%, indicate that both rules provide excel-lent predictions of regime change and duration.

TABLE 1. Contingency table for Rule 1 (a change of re-gime takes place in the orbit after the appearance ofa red star), computed over 40,000 time steps, with 187changes of regime. These numbers correspond to ahit rate (percentage of the forecasts correctly antici-pating the subsequent change or lack of change of re-gime) HR = 91.4%, a threat score or critical successindex TS = 80.3%, and a false-alarm rate, the percent-age of forecasts in which a change of regime was fore-cast but did not occur, FAR = 6.5% (Wilks 1995, pp.238–241).

OBS Yes No Total

FCST

Yes 187 13 200

No 33 299 332

Total 220 312 532

TABLE 2. Contingency table for Rule 2 (fewer than fivered stars in the old regime indicate that the new re-gime will only last three orbits or less, see Fig. 3), com-puted over 40,000 steps. These numbers correspondto HR = 92.0%, TS = 90.0%, and FAR = 2.2%.

OBS Yes No Total

FCST

Yes 134 3 137

No 12 38 50

Total 146 41 187

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of Maryland, and for the extension of support to Erin Evans.This paper was submitted to BAMS in December 2002.

REFERENCESElsner, J., and A. Tsonis, 1992: Nonlinear prediction,

chaos and noise. Bull. Amer. Meteor. Soc., 73, 49–60.Kalnay, E., 2003: Atmospheric Modeling, Data Assimilation

and Predictability. Cambridge University Press, 341 pp.——, M. Peña, S.-C. Yang, and M. Cai, 2002: Breeding and

predictability in coupled Lorenz models. Proc. ECMWFSeminar on Predictability, Reading, U.K., in press.

Lorenz, E. N., 1963: Deterministic non-periodic flow.J. Atmos. Sci., 20, 130–141.

Nese, J., 1989: Quantifying local predictability in phasespace. Physica, 35D, 237–250.

Nicolis, J., G. Mayer-Kress, G. Haubs, 1983: Non-uni-form chaotic dynamics with implications to informa-tion processing. Z. Naturforsch., 38a, 1157–1169.

Palmer, T. N., 1993: Extended-range atmospheric pre-diction and the Lorenz model. Bull. Amer. Meteor.Soc., 74, 49–66.

Sparrow, C., 1982: The Lorenz Equations: Bifurcations,Chaos and Strange Attractors. Springer-Verlag,269 pp.

Toth, Z., and E. Kalnay, 1997: Ensemble forecasting atNCEP and the breeding method. Mon. Wea. Rev.,125, 3297–3318.

Tsonis, A. A., 1992: Chaos: From Theory to Applications.Plenum Press, 274 pp.

Wilks, D. S., 1995: Statistical Methods in the AtmosphericSciences: An Introduction. International GeophysicalSeries, Vol. 59, Academic Press, 464 pp.

THE MAKING OF A METEOROLOGISTA Q&A WITH JOHANNES VERLINDECHICAGO O’HARE INTERNATIONAL AIRPORT JULY 1996–APRIL 2002SUMMER INTERNSHIPS HELPING STUDENTS CHART A CAREER PATHINTERNSHIPS AT TELEVISION AND RADIO STATIONSRISE UNDERGRADUATES FIND THAT REGIME CHANGES IN LORENZ’S MODEL ARE PREDICTABLE