Lack of scaling in global climate models

Transcription

1 INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER J. Phy.: Conden. Matter 14 (2002) PII: S (02) Lack of caling in global climate model D Vjuhin 1,3,RBGovindan 1,3, S Brenner 2, A Bunde 3, S Havlin 1 and H-J Schellnhuber 4 1 Minerva Centre and Department of Phyic, Bar-Ilan Univerity, Ramat-Gan 52900, Irael 2 Department of Geography, Bar-Ilan Univerity, Ramat-Gan 52900, Irael 3 Intitut für Theoretiche Phyik III, Jutu-Liebig-Univerität Gieen, Heinrich-Buff-Ring 16, Gieen, Germany 4 Potdam Intitute for Climate Impact Reearch, D Potdam, Germany govindan@yafit.ph.biu.ac.il Received 9 January 2002 Publihed 22 February 2002 Online at tack.iop.org/jphycm/14/2275 Abtract Detrended fluctuation analyi i ued to tet the performance of global climate model. We tudy the temperature data imulated by even leading model for the greenhoue ga forcing only (GGFO) cenario and tet their ability to reproduce the univeral caling (peritence) law found in the real record for four ite on the globe: (i) New York, (ii) Brooking, (iii) Tahkent and (iv) Saint Peterburg. We find that the model perform quite differently for the four ite and the data imulated by the model lack the univeral peritence found in the oberved data. We alo compare the caling behaviour of thi cenario with that of the control run where the CO 2 concentration i kept contant. Surpriingly, from the caling point of view, the imple control run perform better than the more ophiticated GGFO cenario. Thi comparion indicate that the variation of the greenhoue gae affect not only trend but alo fluctuation. 1. Introduction Indication of weather peritence over month and eaon are known [1]. Uing DFA and wavelet analyi, it ha been hown recently [2, 3] that the variation of the daily temperature from their average value (eaonal trend) exhibit a caling law which i independent of the location of the ite. Indication of uch a peritence law through pectral analyi have alo been found [4, 5]. It wa found that the peritence, characterized by the autocorrelation function C() of temperature variation eparated by day, approximately decay a γ, with roughly the ame correlation exponent γ = 0.7 for the different ite conidered [2, 3]. The range of thi univeral peritence law exceed one decade, and there i no evidence that thi law break down even on longer time cale. Climate model are important ince they are believed to predict the climate change that occur a a reult of anthropogenic interference with atmophere. Here we tudy even /02/ $ IOP Publihing Ltd Printed in the UK 2275

2 2276 D Vjuhin et al tate-of-the-art model: GFDL-R15-a, CSIRO-Mk2, ECHAM4/OPYC3, HADCM3, CGCM1, CCSR/NIES, and NCAR-PCM. Each model ha certain unique characteritic uch a the numerical method, the type of ubgrid-cale parameterization, the patial reolution, and the period of integration. However, all are baed on the ame principle, all have common variable uch a temperature, preure and precipitation and all are coupled to ocean dynamic. A one tet of the validity of the model imulation, we would expect the univeral caling law found in the real obervation to alo be identifiable in the virtual world a repreented by the model output. Thi paper i tructured a follow: in ection 2, we dicu the methodology of fluctuation analyi (FA) and detrended fluctuation analyi (DFA). In ection 3, we conider the record analyi and dicu the reult, and in ection 4 we preent concluion. 2. Methodology 2.1. FA and DFA In order to explain the method, let u conider the monthly mean temperature data T i meaured at a meteorological tation. The index i count the month in the record, {i = 1, 2,...,N}. Figure 1(a) how a portion of the monthly temperature data ( ) for Brooking imulated by the GFDL model. For eliminating periodic, eaonal trend we obtain the deviation of T i, T i = T i T i (ee figure 1(a), olid curve how the climatological annual cycle, T i ), from the average of the monthly temperature T i for each month i, ay January, which ha been obtained by averaging over all year in the temperature erie. Figure 1(b) how the temperature fluctuation T i about the annual average. Rather than calculating C() directly from T i, which i hindered by the poible non-tationaritie in the data and by the finite length of the record, we tudy the fluctuation in the temperature profile, Y n = n i=1 T i in window of ize. To thi end, we divide the profile into K = 2[N/] non-overlapping window of ize (indexed ν) tarting from the beginning a well a from the end. In the imple FA we calculate the quare of the difference of the profile at both end of each egment ν and their average over all egment which give the quare of the fluctuation function F 2 () = 1 K (Y ν Y (ν 1)+1 ) 2. (1) K ν=1 Thi procedure i repeated for different length cale to obtain a caling relation between F()and. For long-range correlated data, F()increae with according to the following power law [6] F 2 () 2α α = 1 γ 2 (2) where α i the fluctuation or caling exponent. If α = 0.5, the ignal i uncorrelated; if α>0.5, there are poitive correlation; if α<0.5, the ignal i anticorrelated. The drawback of FA like the Hurt analyi and the power pectrum analyi i that F()i trongly affected by the preence of trend. To eliminate trend in a ytematic way and thereby learn about them we alo employ everal order of DFA [7 9]. The method ha been applied to a variety of record ranging from DNA equence to atmopheric data through heart-beat time erie, ee e.g. [2, 10, 11]. In DFA, we fit the data of the profile inide each egment (ν) of ize by uing a kth order polynomial function pν k (i), which i called the local trend. The order of the polynomial ued

3 Lack of caling in global climate model (a) 10 T i 0-10 T i (b) i Figure 1. Interpolated monthly temperature data of Brooking imulated by the GFDL model. (a) Solid curve repreent the annual cycle T i for 2 year and the monthly temperature are hown by. (b) Temperature fluctuation around the mean value T i. for the fit fixe the order of DFA. We detrend the profile Y(i) by ubtracting the local trend pν k (i) in each box, and calculate the detrended fluctuation function F ν (i) = Y ν (i) pν k (i). (3) Thu, DFA order k eliminate the polynomial trend of order k from the profile and hence trend of order k 1 from the original ignal. For a given egment of ize, we calculate the DFA mean quare fluctuation function F 2 () = 1 K K ν=1 1 ν i=((ν 1)+1) [F ν (i)] 2 2α. (4) The mean quare fluctuation function obtained from DFA ha the ame caling behaviour a equation (2). It i now quite clear from the above argument that the trend will not be removed in the imple FA method, but they are ytematically removed by different order of DFA. In fact, the pectral method and the Hurt method lack thi ability to ditinguih between trend and correlation. In order to obtain a reliable etimate of the fluctuation exponent, one hould increae the order of DFA until the convergence in the α value i achieved. Thu, by comparing the reult of FA and DFA, one can identify both trend and correlation in the record. 3. Record analyi In thi ection, we apply FA and DFA method to the oberved record at four location in order to corroborate the earlier finding [2]. We then apply the ame method to the data imulated by even leading GCM for the ame four ite.

4 2278 D Vjuhin et al 10 3 New York: (286.00, 40.70) Brooking: (263.22, 44.30) F() F() Saint Peterburg: (30.25, 59.88) Tahkent: (69.22, 41.27) Figure 2. Reult of FA (*) and DFA ( ) for oberved data. In each panel curve from top to bottom repreent FA and DFA1 DFA5. The cale of fluctuation function i arbitrary. Unit of i month. Solid line hown (for comparion) at the bottom of each panel i the theoretical line with a lope Number within parenthee repreent the geophyical co-ordinate of the conidered ite Application to the oberved temperature data Figure 2 how the reult of FA ( ) and DFA ( ) for the oberved data for the four ite New York (USA), Brooking (USA), Saint Peterburg (Ruia) and Tahkent (Uzbekitan). In each panel, curve from top to bottom repreent the reult of FA and DFA1 5. After about 1 year the DFA curve are parallel to each other with a lope of 0.65 ± 0.02 (compare with the line with lope 0.65 hown at bottom). Thi finding i in agreement with earlier reult [2]. In contrat the FA curve for New York (after 1 year), and Brooking (after 2 year) bend upward yielding a higher exponent. Thi indicate the preence of a weak trend, probably due to urban warming, which i eliminated by the econd order and ubequently by higher order of DFA. Thu, it hould be adequate to conider the reult of DFA3 for comparing the caling performance of the global climate model (GCM) Application to the data imulated by GCM Before conidering the reult of the FA and DFA, we will preent ome of the eential detail of GCM. During the pat two to three decade, general circulation model have proven to be an indipenable tool for the tudy of geophyical fluid dynamic ytem. Thee model provide numerical olution of filtered form of the Navier Stoke equation devied for imulating meo-cale to large-cale atmopheric and oceanic dynamic. In addition to the explicitly reolved cale of motion, thee model alo contain parameterization cheme repreentating the o-called ubgrid-cale procee, uch a radiative tranfer, turbulent mixing, boundary layer procee, cumulu convection, precipitation, and gravity wave drag. A radiative cheme,

5 Lack of caling in global climate model 2279 for example, i neceary for imulating the role of variou greenhoue gae uch a CO 2. When run for the global domain, the only boundary condition that need to be pecified are thoe decribing the effect of the urface of the Earth. A typical coupled atmophere ocean (AOGCM) climate model will have a grid pacing of km and vertical layer a compared to weather forecating model which have a grid pacing of 100 km or le and layer. Thi difference in patial reolution i due to the much longer integration time needed for climate model. The difference among the model developed by the leading reearch centre uually lie in the election of the numerical method employed, the choice of the patial reolution, and the detailed formulation of the ubgrid-cale parameterization cheme. Each model i uually ubjected to a rigorou evaluation procedure by it developer including conitency and enitive tet. Model are alo updated by comparing them with the hitorical data. Thee effort have been, however, retricted to traditional time erie analyi of overall characteritic of the entire imulation period. Recently, model inter-comparion have become quite popular. Even in thee inter-comparion project the evaluation are motly baed on the ame diagnotic tool a ued for the individual model aement and relate, therefore, only to mean tate and imple meaure of variability. In general, the traditional method aume that the tatitical propertie of a ignal remain the ame throughout it unfolding period which i not true for the climate record due to the uperimpoed anthropogenic effect. Today, elected reult of tate-of-the-art climate model are readily available through the Internet. Here we conider the monthly temperature record with the greenhoue ga forcing only cenario 5 from even AOGCM: GFDL-R15-a [12], CSIRO-Mk2 [13], ECHAM4/OPYC3 [14], HADCM3 [15], CGCM1 [16], CCSR/NIES [17], NCAR-PCM [18] that are available from the IPCC Data Ditribution Centre [19]. We extracted the data for four repreentative ite around the globe (New York, Brooking, Tahkent and Saint Peterburg). For each model, we elected the four cloet grid point to each ite and bilinearly interpolated the data to the geographical location of the ite. Some of the primary characteritic of the model are lited in table 1. All model are global, include a full uite of ubgrid-cale procee, and explicitly conider ocean dynamic. Note that HADCM3 i the only grid-point AOGCM while the other ix model ue the pectral method. All of the ocean model are grid-point model and do not necearily have the ame patial reolution and grid a the atmopheric model. The period for which the model were run are: GFDL-R15-a ( ), CSIRO Mk2 ( ), ECHAM4/OPYC3 ( ), HADCM3 ( ), CGCM1 ( ), CCSR/NIES ( ) and NCAR/PCM ( ). All the model ued a hitorical level of CO 2 until 1990 and after that they increaed the CO 2 level at a rate of 1% per year. The latter effect hould mainly caue an increae in the trend when we conider the entire record and hould not affect the caling. Therefore, if the model perfectly mimic the real data then they hould yield a caling exponent a found in the oberved record. Reult obtained from the real data (ee figure 2) how that in double logarithmic repreentation, the higher order DFA (DFA2 5) curve are parallel to each other and hence for comparion it i appropriate to ue the reult of DFA3. Reult of DFA3 for F ()/ 1/2 for the data imulated by the even GCM for the four ite New York, Brooking, Saint Peterburg and Tahkent are hown in figure 3. Alo, the line with lope 0.15 (correponding to the real data) i drawn at the bottom of each panel for comparion. It can be readily een from figure 3 that, for a given ite, all the model how 5 In addition to thi cenario there i one more cenario available for all model which i the greenhoue ga plu ulfate aerool forcing.

6 2280 D Vjuhin et al New York Brooking F()/ 1/ 2 GF CS EC HA CG CC NC Saint Peterburg GF CS EC HA CG CC NC Tahkent F()/ 1/ GF CS EC HA CG CC NC GF CS EC HA CG CC NC 10 3 Figure 3. Reult of DFA3 for four ite (New York, Brooking, Saint Peterburg, Tahkent) imulated by the even GCM, GF: GFDL, CS: CSIRO, EC: ECHAM4/OPYC3, HA: HADCM3, CG: CGCM1, CC: CCSR/NIES, NC: NCAR PCM. The curve hown in each panel are the DFA3 fluctuation function divided by 1/2 obtained for different model for the ite conidered in that panel. Theoretical line with lope 0.15 i drawn at the bottom of each panel for comparion. The cale of fluctuation function i arbitrary. Unit of i month. Table 1. Primary characteritic of GCM. Atmopheric Model Origin Type reolution Coupling GFDL-R15a GFDL (Princeton) Spectral R15L9 Flux correction CSIRO-Mk2 CSIRO (Autralia) Spectral R21L9 Flux correction ECHAM4/OPYC3 DKRZ (Hamburg) Spectral T4L19 Flux correction HADCM3 Hadley Centre (UK) Grid point M96N73L19 Flux conervation CGCM1 CCC (Canada) Spectral T32L10 Flux correction CCSR/NIES CCSR (Japan) Spectral T21L20 Flux correction NCAR-PCM NCAR (Colorado) Spectral T42L18 Flux conervation wide difference in caling behaviour. Some of the model how uncorrelated behaviour after a few year (ee figure 3 ECHAM4/OPYC3 model and table 2) which implie lo of memory. For New York, the HADLEY model yield an exponent of 0.64 which i cloe to the oberved data. None of the model perform well for Tahkent, Saint Peterburg and Brooking. Table 2 ummarize the exponent obtained for the four ite imulated by different model. It can een from table 2 that mot of the exponent value are in the range of , although for ome of the ite the exponent are cloer to the oberved data. Alo, the mean and tandard deviation value of thee exponent are given in table 2 (in an ideal ituation the mean and tandard deviation of the caling exponent obtained from the model hould be 0.65 and zero, repectively), which indicate the inconitency of the model.

7 Lack of caling in global climate model CSIRO Mk2: ECHAM4/OPYC3: F() CGCM1: CCSR/NIES: F() Figure 4. Comparion of fluctuation function of GGFO cenario and CR for four GCM. Curve from top to bottom in each panel how fluctuation function of FA ( ) and DFA1 DFA5 ( ). Fluctuation function of GGFO cenario are hown by olid ymbol while thoe of CR are hown by open ymbol. Theoretical line with lope 0.5 (bottom) and 0.65 (top) are drawn for comparion. The cale of fluctuation function i arbitrary. Unit of i month. Table 2. The fluctuation exponent α obtained for different model for different citie. Brooking New York St Peterburg Tahkent Mean Std. GFDL-R15a CSIRO-Mk ECHAM4/OPYC HADCM CGCM CCSR/NIES NCAR PCM Next, we compare the caling behaviour of the temperature data obtained from the GGFO cenario with that of the control run (CR). Figure 4 how repreentative reult for Tahkent for the four model: (i) CSIRO-Mk2, (ii) ECHAM4/OPYC3, (iii) CGCM1 and (iv) CCSR/NIES. The other three GCM do not have data for the period dicued in figure 4 for CR and hence we have not conidered them for the comparion. Figure 4 how the fluctuation function of FA ( ) and DFA1 5 ( ) for the GGFO cenario (olid ymbol) and CR (open ymbol). Since trend are not eliminated in FA, fluctuation function of FA for CR yield lower exponent when compared to thoe obtained for the GGFO cenario, a expected. For the cae of CR, the

8 2282 D Vjuhin et al fluctuation function of DFA1 5 how a caling exponent cloe to 0.59 for CSIRO-Mk2, 0.65 for ECHAM4/OPYC3 and CGCM1, and about 0.6 for CCSR/NIES, which are le far from the univeral exponent found in the oberved data (α = 0.65). Thi how that the addition of increaed CO 2 in the model caue the imulated reult to loe the oberved caling (for comparion ee the fluctuation exponent for Tahkent for the ame model in table 2). 4. Concluion None of the GCM for the GGFO cenario how caling performance cloe to the oberved data. The lower exponent value obtained for the GGFO cenario how that the model lack the peritence found in the oberved data. In addition, the model diplay a wide range of caling performance. Alo, the comparion of the caling exponent of CR veru GGFO cenario how that the gradual addition of CO 2 make the model loe their memory during the coure of imulation. Even though the addition of CO 2 in the model are baed on fair aumption, they are not being done in a proper way. The introduction of a gradual increae of CO 2 caue ome other ide effect apart from it uual role of increaing the temperature, which caue the model to lack the univeral caling behaviour. A careful analyi of the methodology of the addition of CO 2 may certainly improve the performance of the model. Reference [1] Shukla J 1998 Science [2] Kocielny-Bunde E et al 1998 Phy. Rev. Lett [3] Kocielny-Bunde E, Bunde A, Havlin S and Goldreich Y 1996 Phyica A [4] Talkner P and Webber R O 2000 Phy. Rev. E [5] Pelletier J D and Turcotte D L 1997 J. Hydrol [6] Bunde A and Havlin S 1995 Fractal in Science (New York: Springer) [7] Peng C K et al 1994 Phy. Rev. E [8] Kantelhardt J W et al 2001 Phyica A [9] Hu K et al 2001 Phy. Rev. E [10] Bunde A et al 2000 Phy. Rev. Lett [11] Buldyrev S V et al 1995 Phy. Rev. E [12] Dixon K W and Lanzante J 1999 Geophy. Re. Lett [13] Gordon H B and OFarrell S P 1997 Mon. Weather. Rev [14] Chritoph M et al 2000 J. Clim [15] Pope V D et al 2000 Clim. Dyn. 123 [16] Flato G M et al 2000 Clim. Dyn [17] Emori S et al 1999 J. Meteorol. Soc. Jpn [18] Wahington W M et al 2000 Clim. Dyn [19] index.html