Fractal dimensions of convective clouds around Delhi

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1 , j Indian Jurnal f Radi & Space Physics Vl. 27, February 1998, pp.i-6 Fractal dimensins f cnvective cluds arund Delhi Kaushar AIi, R N Chatterjee, Prem Prakash & P C S Devara Indian Institute f Trpical Meterlgy, Pune B RD and Gupta Department f Gephysics, Banaras Hindu University, Varanasi Received 19 March 1997; revised received 13 Octber 1997 Cluds exhibit fractal structure ver wide ranges f scale. Fllwing the methd suggested by Lvejy [Science (USA), 216 (1982) 185], fractal analysis f cnvective cluds, bserved within 100 km arund Delhi during the mnsn seasn (July-September) f 1977, has been carried ut. The study is based n the radar bservatins cllected with a high pwer 3-cm weather radar. A ttal number f 2568 radar eches f cnvective cluds having areas between 4 and 7420 km 2 were analysed fr studying the relatinship between their perimeters (P) and areas (A). The analysis shwed that the relatinship between the abve tw ech parameters can well be represented by a frmula P_ADI1, where D is described as the fr-actal dimensin f the clud ech perimeter. Hwever, it was fbund that the tw fractal dimensins f the perimeters are required t describe the abve perimeter-area relatinship applicable fr the entire range f clud ech area. That is, D =1.34 fr clud eches having areas up t 600 km 2 and D = 1.79 fr eches having areas greater than 600 km'. 1 Intrductin Cluds are self-similar fractals because they shw a cnsistency in the relatin between their areas (A) and perimeters (P) ver a wide range f length scale':'. The cncept f fractals prvides a systematic apprach tward the quantitative characterizatin f the fragmented structures like cluds. It has appreciably cntributed t the understanding f the turbulent diffusin in the atmsphere and the physical mechanisms f the frmatin f cluds':". A dimensinal analysis f such fractal areas using A-P relatien helps in understanding the irregularity r degree f cntrtin f the perimeters enclsing the areas. The term "clud" in the present cntext refers t the radardetermined clud, i.e. the clud ech as seen n PPI scpe f-the radar. Accrding t Lvejy I, the A-P relatin may be expressed as p_adiz, where D is the fractal dimensin f perimeter f cluds which characterizes the degree f cntrtin f the perimeter enclsing the hrizntal area f the cluds. If the shape f the clud area is smth, the length f the perimeter t enclse the area will be minimum and the value f D will be I, the dimensin f a line. The A-P relatin then takes the frm P_AII2. As the clud area becmes mre and mre irregular the perimeter required t enclse the area will be mre and mre cntrted, increasing thereby the length f the perimeter. This cntrtin accunts fr the fractinal value f the dimensin f the perimeter. That is, the fractal dimensin is a functin f irregularity f the surfaces. The upper limit f the cntrtin f perimeters will tend t duble back n itself, filling the area in the plane. This leads the abve relatin t lk as P - A and D appraches the value 2. In ur earlier wrk' n the fractal dimensin f cnvective cluds arund Delhi, we fund that the pre-mnsn cnvective clud eches shw cnsistency in the relatin between area and perimeter with area ranging frm 4 t 984 km 2 and fractal dimensin being Als, it may be added that ut f 397 clud eches sampled during the perid f that study, mst f them related t smaller size ranges.

2 INDIAN J RADIO & SPACE PHYS, FEBRUARY 1998 Table I-Radar characteristics Table 2-Distributin f ech areas Wavelength Peak pwer transmitted Pulse length Minimum detectable signal Pulse repetitin

2 2 INDIAN J RADIO & SPACE PHYS, FEBRUARY 1998 Table I-Radar characteristics Table 2-Distributin f ech areas Wavelength Peak pwer transmitted Pulse length Minimum detectable signal Pulse repetitin frequency Hrizntal and vertical beamwidth 3.2cm 250kW I IlS -90dBm 300Hz 1.2 The present study which is a cntinuatin f the abve wrk deals with the size distributin and the fractal dimensin f cnvective clud eches bserved within 100 km arund Delhi during mnsn seasn (July-September) f This study was taken up als t investigate whether there exists any marked difference between the fractal nature f cnvective cluds in summer seasn and that in mnsn seasn. 2 Data used and methd f analysis The study is based n hurly radar bservatins f precipitatin eches frm cnvective cluds within 100 km arund Delhi, made during the mnsn (July-September) seasn f An X- band Japanese radar f type NMD-451A has been used fr this purpse. The characteristics f the radar set are given in Table 1. The bservatins were mainly taken frm 1000 t ST (0430 UTe t 1130 UTC). Hurly phtgraphs f PPI bservatins btained at an antenna elevatin angle f were srted ut. Frm these phtgraphs, precipitatin eches frm cnvective cluds were selected and their areas and perimeters were measured. In all 2568 ech cases were sampled during the perid f study. The perimeters f the individual clud eches were measured with a reslutin f 2 km and the areas were measured with a reslutin f 4 km 2 by cunting the number f 2 km x 2 km grids cvered by the individual clud eches. The range f radar bservatins fr the present study has been cnfined up t 100 km nly. This is due t the fact that fr larger ranges errrs may be intrduced in the bservatins f the clud dimensins due t the effect f radar antenna beamwidth and earth's curvature, and hence the cmputatin f fractal dimensin may lead t sme errneus results. 3 Results and discussin 3.1 Size distributin f cnvective eches Befre we prceed t discuss the fractal statis- Size distributin km 2 : ~4200 Ttal N. f cases Percentage f cases tics f the radar eches frm cnvective cluds sampled fr the present study, it is imprtant t discuss first their size distributin. Fr this purpse, eches have been classified int different grups accrding t their areas as shwn in Table 2. It may be seen that ut f 2568 clud cases, areas f abut 60 per cent f them were less than 'r equal t 16 krrr', areas f as much as 87 per cent were less than r equal t 60 krrr', and areas f nly 2.9 per cent exceeded 600 km", Area f the smallest clud ech was 4 km 2 and that f the largest was 7420 krrr'. 3.2 Area-perimeter relatinship and fractal dimensin f cnvective cluds Figure 1 shws a scatter plt f lgarithm f perimeter (P) versus lgarithm f area (A) fr the 2568 cnvective eches. It may be mentined that a large number f eches have been fund t have the same perimeter fr a particular area f the eches mainly in the small ech range. As such, the number f pints pltted in the scatter diagram is less than the actual number f clud eches sampled. The trend analysis suggests that the pints in the scatter diagram can be fitted by tw straight lines. As such, the pints were fitted t tw straight lines by the methd f least squares. Line 1 is fr clud eches with areas less than r equal t 600 km 2 (cvering 97 per cent f ttal ech ppulatin) and line 2 is fr the eches with areas larger than 600 km 2 (cvering 3 per cent f the ttal ppulatin). Equatins fr the tw regressin lines, 1 and 2, are given by

3 KAUSHAR ALl et al.: FRACfAL DIMENSIONS OF CONVECflVE CLOUDS AROUND DELHI 3 LINE I: P = 2.49 AO.668 LINE 2: P = 0.84 AO ~-a: E - IO'i. wa. w~w::e fo' 10 10' AREA (km2) Fig. I--I..g-Ig plt f the perimeters f radar eches f cnvective cluds as a functin f their areas. Tw regressin lines have been fitted t the pints representing area (..4)- perimeter (P) relatinships f the eches (Line I is fr eches f areas up t 600 km2, Line 2 is fr eches f areas greater than 600 Jan2; crrespnding regressin frmulae are als indicated in the figure). lg P = lg A (1) lg P = lg A (2) Frm Eqs (1) and (2) and the relatin P=PADI2, where P is the cnstant f prprtinality, least square estimates f the fractal dimensin (D) f perimeters f cnvective eches have been deduced fr the tw grups as (a) Fr cluds with ech areas ~ 600 km2 D = 2 x slpe f the line 1 = 2 x = 1.34 (3) lg P = r P = 2.94 (4) (b) Fr cluds with ech areas> 600 km2 D = 2 x slpe f the line 2 = 2 x = 1.79 (5) lg P = r P = 0.84 (6) The present results indicate that the smaller clud eches (area ~ 600 km2) which ccur in majrity are less irregular in shape than larger clud eches. It is assumed that the cluds are selfsimilar fractals, i.e. with n characteristic length scale. In ther wrds, they are statistically invariant under certain transfrmatins f scale. The

I'll 4 INDIAN J RADIO & SPACE PHYS, FEBRUARY 1998 break in the slpe, as btained in ur study f A-P relatin f cnvective clud eches, reveals that the assumptin f scale invariance ver the whle gamut f

4 I'll 4 INDIAN J RADIO & SPACE PHYS, FEBRUARY 1998 break in the slpe, as btained in ur study f A-P relatin f cnvective clud eches, reveals that the assumptin f scale invariance ver the whle gamut f clud size can be discarded. Cahalan and Jseph4 have als fitted tw lines thrugh the pints f A-P relatin, but the clud ech area at which the break bserved by them (3.14 km2) is quite different frm that nticed in the present study (600 km2). Rather, scale invariance appears t be an excellent assumptin fr the smaller and larger clud grups separately. Althugh, this prperty has been fund t hld gd fr larger cluds t, it is difficult t emphasize the result fr the reasn that the data pints are less in number. In the present study the gruping f smaller and larger cluds has been dne n the basis f the trend f scatter plt in Fig. 1. There is a need t search sme fundamental criteria fr such grupings which must serve in evaluating the break pint in the slpe f A-P relatin. Als, there is a need t further investigate the upper limit f larger cluds predicted by line 2. In ur earlier study f fractal dimensins, the data pints representing cnvective ech areas greater than 600 km2 were meagre (nly 0.6 per cent f the ttal ppulatin f 397) which aligned t the same line trend. As such, the prperty f scale invariance appeared t exist ver the whle range f the bserved clud area. This is cnsistent with the findings f Jain2 wh suggested that the assumptin f scale invariance hlds gd fr cnvective clud ech areas even up t 5350 km2 The fractal dimensins btained in ur previus and present studies depict that, in general, the wiggliness in perimeters f summer clud eches lies in between the wiggliness in perimeters f smaller and larger mnsn clud eches. The fractal dimensin btained in the present study fr smaller cluds (1.34) is in gd agreement with that bserved by Lvejyl (1.35) and Jain2 (1.30). Visual examinatin f the radar eches indicates that mst f the small cnvective eches which have happened t ccur in islatin in the clud field lk less irregular in shape and have unifrm brightness thrughut. Hwever, many ther clud eches with areas limited t 600 km2 have been fund t shw several regins f prnunced and diffused brightness. This suggests clud merging as a frmatin prcess fr such cluds. Because f the merging f several clud eches, the cntact pints exhibiting diffused brightness there give rise t a mre cntrted perimeter enclsing the whle area f the s frmed cluds. Such cluds have als been fund t ccur in the same scaling regime as represented by line 1 f Fig.l. This implies that the prcess f clud merging t frm larger cluds des nt alter the shape f smaller cluds but nly magnifies it. The larger clud eches with areas greater than 600 km2 are prbably frmed when cnvective cells are accmpanied by stratifrm eches (as inferred frm the presence f melting band). Chatterjee et al.6, in their study f cnvective cluds within 100 km arund Delhi during mnsn seasn f , have shwn that the cluds having areas up t 600 km2 are frmed either by the prcess f randm entrainment f envirnmental air r by merging f smaller cnvective elements and the larger cluds are the result f stable messcale extensin f the stratifrm eches accmpanying the cnvective cells. This feature explains the higher fractal dimensin fr the larger clud eches. Cahalan and Jseph4 have suggested that the smaller cluds develp by the bundary layer cnvectin whil?the larger cluds by clud merging. They have als suggested that, perhaps, the cut-ff scale ccurring in the A-P relatin can be determined by the thickness f the bundary layer, and change in fractal dimensin can be related t a change in three-dimensinal turbulence in the bundary layer t tw-dimensinal turbulence invlving clud mergers in the free atmsphere. But accrding t their cnceptin smaller cluds are thse cluds which have diameters less than 2 km (crrespnding t an area f 3.14 km2). In ur wrk, smaller cluds range frm an area f 4 km2 t 600 km2 Thus, the study suggests an additinal regime which may range frm an area in the vicinity f 4 km2 dwn t the limit f the reslutin. There is clearly a need fr mre research n the factrs determining the size f the cnvective cells, and mre generally n the relatin f different scaling regimes t different physical prcesses in the atmsphere. 3.3 Fractal dimensin f individual cnvective eches Using the relatin P = pam, where P = 2.94 fr cnvective eches with areas:::;;600km2 and P 1 11 'ill I 1'1' II 'i' "'''''''1 'I' 1 1

5 KAUSHAR ALl et at.: FRACTAL DIMENSIONS OF CONVECTIVE CLOUDS AROUND DELHI 5 50 TOTAL NUMBER OF CLOUDS' TOTAL NUMBER OF CLOUDS = r IAO 144 1: FRACTAL DIMENSION (OJ Fig. 2-Percentage frequency distributin f the fractal dimensin (D) f'perimeters f the radar eches frm cnvective cluds f areas up t 600 km2 = 0.84 fr cnvective eches with areas> 600 km2, fractal dimensins f individual cnvective eches have been determined in rder t study their spread abut the least square estimate f fractal dimensin D = 1.34 and 1.79 respectively fr the tw grups f eches. Frequency distributin f the values f D in different class intervals fr bth the grups is shwn in the frms f histgrams in Figs 2 and 3 respectively. It may be seen that fr the grup f eches with areas limited up t 600 km2 (Fig. 2) the range f D values lies between 1.16 and 1.68 with standard deviatin f Their mean, median and mdal values are 1.34, 1.33 and 1.34 respectively. The spread f D values frm the nrmal distributin is very small and the deviatin shws psitive skewness. But when the result is cmpared with the standard Gaussian and lg-nrmal distributins, it came ut t be highly significant (at 0.1 per cent level) in bth the cases, i.e. the result is highly deviated frm these distributins. Out f the 2494 cnvective eches in this grup, as many as 2170 eches (i.e. 87 per cent) exhibited D = 1.34 ± Fr the grup f clud eches with areas greater than 600 km2, range f D values varied between 1.74 and 1.84 with standard deviatin f Thi.s implies less dispersin f D values arund the mean. Their mean, median and mdal values are 1.79, 1.80 and 1.80 respectively. This shws a deviatin in negative directin frm nrmality. Again the Gaussian and lg-nrmal fittings thrugh this spread shwed the results t be highly significant (at 0.1 per cent level). Out f the 74 cnvective eches in this grup, 62 eches (i.e. 84 ~ 40 lj) ILl ~ 30 ILl IJ :: ILl m ~ ;::) 10 z Fig. 3-Same as Fig. 2 but fr areas greater than 600 km2 per cent) exhibited D = 1.79 ± 0.03 and 100 per cent f the eches exhibited D = 1.79 ± Summary and cnclusins The area-perimeter relatinship and fractal dimensins f the perimeters f cnvective cluds arund Delhi have been studied using the radar data f precipitating cluds cllected during the mnsn (July-September) seasn f A ttal number f 2568 clud cases with areas varying between 4 and 7420 km2 have been analysed. A scatter plt f lgarithm f perimeter (P) versus lgarithm f area (A), wherein the pints are fitted with tw straight lines by emplying the methd f least squares, suggests that tw fractal dimensins are needed t describe the A -P relatinship fr cnvective cluds ccurring in Delhi regin during mnsn seasn, i.e. D = 1.34 fr cnvective eches with areas ~ 600 km2 (cnstituting abut 97 per cent f the ttal eches) and D=1.79 fr the larger areas (> 600 km2) f mre cmplex in structure. Visual inspectin f radar-determined cluds alng with the analysis f bservatins indicates that the prcess f frmatin f cluds having areas up t 600 km2 invlves merging f smaller clud cells. In this prcess the size f the cluds appears as magnified versin f the smaller clud cells and their structure in the hrizntal plane remains almst the same. The break in the slpe f the scatter plt. (Fig. I) is a clear indicatr f the existence f sme different mechanism fr the cluds grwn t be larger

6 6 INDIAN J RADIO & SPACE PHYS, FEBRUARY 1998 than 600 km2 in area, Frmatin f such larger cluds culd be due t the cnvective cells accmpanied by the stratifrm eches. This feature is cnsidered fr the large value f the fractal dimensin fr the bserved larger cluds, The present study pints ut that ne mre scaling regime exists in the fractal structure f cluds ranging frm an area in the prximity f 4 km2 dwn t the limit f reslutin. It may be mentined here that in ur previus study f the structure f cnvective cluds f summer seasn, nly ne fractal dimensin (D= 1.42) was needed t describe the lg (area)-lg (perimeter:) relatinship. This was due t the reasn that ut f the ttal cnvective clud eches sampled in the summer seasn percentage f large eches was very small. Areas f nly 0.6 per cent f the ttal ech ppulatin f 397 exceeded 600 km2 in the summer seasn as cmpared t 2.9 per cent in the mnsn seasn. Maximum ech area bserved in the summer seasn was 984 km2 whereas in the mnsn seasn it has been fund t be 7420 km2., Acknwledgements The authrs are grateful t the Directr, IITM, Pune, Dr A S R Murty and Dr R K Kapr fr their valuable suggestins and encuragement. Thanks are als due t the annymus referees fr their useful suggestins / remarks n the riginal manuscript fthe paper. References I Lvejy S, Science (USA), 216 (1982) Jain P S, Mausam (India), 40 (1989) Hentschel H G E & Prcaccia I, Phys Rev A (USA), 29 (1984) Cahalan R F & Jseph J H, Mn Weather Rev (USA), 117 (1989) Chatterjee R N, Prakash P & Ali K, Mausam (India), 43 (1992) Chatterjee R N, Ali K & Prakash P, Indian J Radi & Space Phys, 23 (1994) 189. II I,:

, which yields. where z1. and z2

, which yields. where z1. and z2 The Gaussian r Nrmal PDF, Page 1 The Gaussian r Nrmal Prbability Density Functin Authr: Jhn M Cimbala, Penn State University Latest revisin: 11 September 13 The Gaussian r Nrmal Prbability Density Functin