FORMULA FOR SPIRAL CLOUD-RAIN BANDS OF A TROPICAL CYCLONE. Boris S. Yurchak* University of Maryland, Baltimore County, Greenbelt, Maryland =, (2)

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1 PF.3 FORMULA FOR SPIRAL CLOUD-RAIN BANDS OF A TROPICAL CYCLONE Bois S. Yuchak* Univesity of Mayland, Baltimoe County, Geenbelt, Mayland. INTRODUCTION Spial cloud-ain bands (SCRB) ae some of the most distinguishing featues inheent in satellite and ada images of topical cyclones (TC). The finding of eliable indicatos of the TC s intensity and its cente location is one of the fields in SCRB study. The kind of function that is most fitting fo desciption of an SCRB is an impotant point in such eseach. This poblem is diectly connected with the mechanism of SCRB fomation and evolution. In the cuent wok, the simplest hypothesis of entapment of cloud masses was used. The coe of this hypothesis is an assumption that clouds ae involved by the spial wind steamlines flown into the cente of a cyclone. u S α µ V v φ O. EQUATION FOR WIND STREAMLINE IN TC Accoding to Mamedov and Pavlov (974), the steamline equation in a pola coodinate system can be witten as d ϕ = u v d, () whee is a adius-vecto, ϕ is an angle between adius vecto and the positive diection of X, and u and v ae tangential (to cicula isoba) and adial components of ai paticle motion, espectively. One of the main paametes of SCRB is the so-called cossing angle which is an angle between the tangent to the steamline (cloud tack) and the tangent to the cicula isoba at a paticula point (Fig.). It is known fom many expeimental obsevations, that the cossing angle deceases with the decease of spial adius as the point is coming to the pole of a spial. It also occus at a fixed distance, due to cyclone intensity incease (Repot WMO (985), Raghavan (3)). Depending on elations among the paametes pesented in the steamline equation (), a solution of this equation will be some kind of a spial. In paticulaly, the logaithmic spial R exp( ϕ tgα) =, () *Coesponding autho addess: Bois S. Yuchak Univ. of Mayland, Baltimoe County, GEST, Geenbelt, MD, , yubois@umbc.edu Fig.. Deivation of the steamline equation fo a TC. () wind steamline; () cicula isoba; O is the TC cente; S is an abitay point within a wind steamline; V is the tangential wind speed at the point S; u and v ae tangential (to cicula isoba) and adial components of the tangential wind, espectively; α and µ ae the cossing angle and the inflow angle, espectively; φ is the pola angle; and is a adius-vecto. whee R is cyclone adius (it is estimated by adius of the last cicle isoba of the TC system), ϕ is spial s pola angle (ϕ = at =R), α is cossing angle, means that adial and tangential wind components have the same law of adius dependence (the cossing angle is constant): v/u=tgα=const. Fom the known expeimental data, pointed in Weathefod and Gay (988), fo example, it does not point out that the above conditions eally might be in TC. The logaithmic spial expession can be pesented also as a dependence of pola angle on adius nomalized by R: ϕ = ctgα ln (3) R In the modified logaithmic spial (Anthes (98)): ln m = ϕ tg ( ) R α, (4) whee: tg = m ( ) tgα tg tgα( ) α, α =,

2 m is maximal wind adius and α is peipheal cossing angle, it is taken into account empiically that the cossing angle deceases as the point appoaches the cente of TC. But it does not depend on TC intensity as well. Thus, these models of the cossing angle change have adjusting and empiical style, and the application of such o othe spial fom is not substantiated. 3. STREAMLINE EQUATION TAKING INTO ACCOUNT THE CHANGES OF THE INFLOW ANGLE AS A FUNCTION OF TC RADIUS Steady-state ai motion in the cyclone causes the fiction foce = kv, whee k is a fiction F f facto. In accodance with Gualnik et al. (97), the deviation angle to the ight fom the gadient foce diected to the cyclone cente (Fig. ) is given by an expession l V ( ) tgµ( ) = +, (5) k k whee l = ω φ sin is the Coiolis paamete (ω is the angula speed of the Eath otation, φ is the altitude) and V is the modulus of paticle s speed vecto. As follows fom the Fig., the cossing angle and inflow angle ae connected by a elationship π α + µ = (6) whence k k tgα ( ) = ctgµ ( ) = = (7) V ( ) l V ( ) l + + l The values of the cossing angle ae povided in Table fo diffeent ations between the Coiolis V ( ) paamete and paamete. Table. The cossing angle values depending on elationship between the paametes of fomula (7) V ( ) ~ 5 5 l α() ~ In calculation of Table, the Coiolis paamete was chosen fo latitude of 9, i.e. 5 l = ω sinφ = 4,8 s, and fiction facto 5 k 8 s. The data of Table show that the ange of the cossing angle is fom to 4 degees, which is in accodance with the values obseved expeimentally (Repot WMO (985)). Because the incease in the atio of wind speed to adius causes the decease in the cossing angle, it is possible to conclude that the cuent model of the cossing angle change is matched qualitatively to the expeimental data (the cossing angle deceases when the point appoaches the cente of a TC and due to TC intensity incease). Solution of the equation () with u v = tgµ, whee tgµ is defined by equation (5), and the speed of wind changes with cyclone adius is in accodance with the powe low (Anthes (98)) m V ( ) = Vm (8) is deived by Yuchak (7) and has a fom: ϕ = A B ln y n y, (9) + whee y = is a elative adius, n ym l m A =, B =, ym =, τ =. kτ ( n +) k V m Equation (9) can be epesented in the exponent-logaithmic fom as well: ( n+ ) ln y ϕ = A { e } B ln y (9а) 3. CROSSING ANGLE OF THE HYPERBOLIC- LOGARITHMIC SPIRAL It is easy to show based on (7) that, in contast to the logaithmic spial, the cossing angle of HLS depends on TC adius: whee k tg( α ( )) = () n+ l m + Ro Vm Ro = is the Rossby numbe. l m Expession () can also be witten as a function of HLS paametes: tg( α ( y)) = n + () B + A n+ y As detailed in (), paticulaly, the coefficient A has a notably geate weight than the coefficient B as the point appoaches the cyclone cente (with deceasing y). This featue, as it will be shown below, is the facto which govens mainly the diffeence between HLS and the logaithmic spial within the cental pat of TC. n

3 4. COMPARISON OF THE LOGARITHMIC AND HYPERBOLIC-LOGARITHMIC SPIRALS Fo illustation, seveal HLSs in pola and Catesian semi-logaithmic (φ-lny) coodinates ae shown in Fig. with paametes: n=.5, V m =4 m/s and 8 m/s, m =4 km, o =4 km, and k=l. At that А=4.374 and А=8.75 espectively and B=. It is easy to show that the logaithmic spial with the most applicable cossing angle of 8 (Willoughby (978)) is the specific case of the HLS with A = and B = This spial is also shown in Fig. fo compaison. The incease of the facto A is due to inceasing TC intensity unde a constant exponent and leads to the ounding of the spial (deceasing of cossing angle). This is in accodance with () as well. It is possible to show also that the change by twice of the facto B at the logaithmic component of the HLS and the exponent n pactically does not impact the shape of the HLS. With A=, the HLS degeneates to the classical logaithmic spial whee tgα = B. Main diffeence between HLS and the logaithmic spial is indicated in the cental pat of a TC. The logaithmic spial goes to the cente much faste then the HLS. The HLS seems like cuve aound the TC s eye. At that, the oundness of this culing is so much the bette as the intensity is highe. 5. SOME EXAMPLES OF APPROXIMATION OF SPIRAL CLOUD-RAIN BANDS IN A TROPICAL CYCLONE BY A HYPERBOLIC- LOGARITHMIC SPIRAL To illustate the pactical applicability of the poposed way of SCRB desciption, the appoximation of spial bands of satellite and ada images of TCs was conducted fo seveal examples. To calculate the HLS and the logaithmic spial paametes, the least-squaes method was applied. An appoximation was done in accodance with a apid design scheme unde the assumption of the constant exponent n equal to.5. The pocessing esults of TC Mitag (UK MetOffice, achive of TC images: e/images.html) and ada images of TC Iving (Yuchak (997)) ae povided in Figs.3-5 and Table. Othe examples of HLS application fo SCRB appoximation and fo a moe exact location of a TC cente ae povided in Yuchak (7)..8.6 HLS_A=4.375 (Vm=4 m/s), B=, n=.5.4. HLS_A=8.75 (Vm=8m/s), B=, n= Log_8 (A=, B=3.77) lny Log_8 (A=, B=3.77) HLS_A=4.375, B=, n=.5 HLS_A=8.75, B=, n=.5 Fig.. Hypebolic-logaithmic spial in pola (top plot) and semi-logaithmic (bottom plot) coodinates with diffeent paametes. Log_8 is the logaithmic spial with 8 cossing angle pola angle, ad lny Iving_33 HLS Log_spial Fig.3. Example of compaative appoximation of the pincipal spial band of TC Iving at 3:3 (7/3/89) by logaithmic spial and HLS. Image at the top is oiginal ada image, bottom plot is spial band points and its appoximation by HLS and logaithmic cuves in semi-logaithmic coodinates. Logaithmic spial is denoted by thin line on ada image and by dashed line on appoximation plot (bottom). Paametes of HLS ae A=.5, B=-.57. Paamete of the logaithmic spial is B L = Fi, ad

4 Fig.4. Appoximation of the pincipal cloud-ain band of TC Iving at 4:3am (7/4/89) by two tuns of HLS (bold line) and the logaithmic spial (thin line). Main diffeence between the spials is obseved close to the cente of TC. Paametes of the spials ae listed in Table. Fig.5. Appoximation of thee spial cloud-ain bands of TC Mitag by one tun of logaithmic (thin line) and HLS (bold line) spials. Top left pictue is an oiginal image (souce: UK MetOffice]. Edited spial cloud-ain bands ae annotated by numbes: is pincipal band, and 3 ae seconday bands. Paametes of spials ae listed in Table.

5 Table. Results of appoximation of the spial cloud-ain bands by hypebolic-logaithmic and logaithmic spials of seveal examples of satellite and ada measuements of TCs TC name, Date, (Refeence) Numbe of a spial Paametes of the HLS A σ A B σ B Residual vaiance B L Paametes of the logaithmic spial σ BL Residual vaiance Residual vaiance atio MITAG, 3/5/ GMS-5 (UK MetOffice) IRVING, 7/4/989, 4:3am, MRL-5 (Yuchak (997)) SUMMARY In this pape, a physically substantiated fomula fo spial cloud-ain bands of TC based on the physical appoach utilization of the law of inflow angle dependence in TC on its adius is pesented. The main featue of the HLS is the dependence of its coefficients on the physical paametes of TC and the envionment. The hypebolic-logaithmic spial obtained includes the known logaithmic spial as a specific case. Main advantage of the HLS ove the empiical logaithmic appoximation is in moe exact desciption of the cental pat of a TC s spial cloud-ain complex. Thee is a base to popose that the obtained fomula might be used also fo appoximation of steamline by pocessing the ada wind field measuement data (Tuttle and Gall (999)) to locate moe exactly the TC s eye cente position. 7. REFERENCES R. Anthes, Topical Cyclones. Thei Evolution, Stuctue and Effects. AMS, Meteoological Monogaphs, No. 4, 98. I. Gualnik, Dubinskii, G.P., and S. V. Mamikonova, The Meteoology. Handbook fo Univesities, Gidometeoizdat, Leningad, 97 (in Russian). S. Raghavan, 3, Rada Meteoology. Kluwe Academic Publishes, 3. Repot of the Semina on the Application of Rada Data to Topical Cyclone Foecasting. Topical Cyclone Pogamme, Repot No. TCP- 9, Bangkok, Thailand Nov. Dec., 983, WMO, Geneva, Switzeland, 985. Weathefod, C.L., and W. M. Gay, 988: Typhoon Stuctue as Revealed by Aicaft Reconnaissance. Pat I: Data Analysis and Climatology. Mon. Wea. Rev., 6, Willoughby, H.E., 978: A Possible Mechanism fo the Fomation of Huicane Rainbands. J. Atmos. Sci., 35, Tuttle, J., and R.Gall, 999: A single-rada Technique fo Estimating the Winds in Topical Cyclones. BAMS, vol. 8, No. 4, Yuchak, B.S., 997: Rada Study of the Eye of Topical Cyclone Iving Passing ove the Gulf of Tonkin. Russian Meteoology and Hydology, No.6, 7-3 (in Russian). Yuchak, B.S., 7: Desciption of Cloud-Rain Bands in a Topical Cyclone by a Hypebolic- Logaithmic Spial. Russian Meteoology and Hydology, vol. 3, No., 8-8. E. Mamedov and N.I.Pavlov, Typhoons. Gidometeoizdat, Leningad, 974 (in Russian).