Power-Law Behavior of Power Spectra in Low Prandtl Number Rayleigh-Bénard Convection

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1 Power-Lw Behvior of Power Spectr in Low Prndtl umber Ryleigh-Bénrd Convection M. R. Pul nd M. C. Cross Deprtment of Physics, Cliforni Institute of Technology , Psden, Cliforni 9115 P. F. Fischer Mthemtics nd Computer Science Division, Argonne tionl Lbortory, Argonne, Illinois H. S. Greenside Deprtment of Computer Science nd Deprtment of Physics, Duke University, Durhm, orth Crolin 7706 (Dted: My 11, 001) The origin of the power-lw decy mesured in the power spectr of low Prndtl number Ryleigh-Bénrd convection ner the onset of chos is ddressed using long time numericl simultions of the three-dimensionl Boussinesq equtions in cylindricl domins. The powerlw is found to rise from qusi-discontinuous chnges in the slope of the time series of the het trnsport ssocited with the nucletion of disloction pirs nd roll pinch-off events. For lrger frequencies, the power spectr decy exponentilly s expected for time continuous deterministic dynmics. PACS numbers: r,47.5.+j,47.0.Bp,47.7.Te Significnt insight into the onset of chotic dynmics in fluid systems, nd continuum systems in generl, hs been gined from cryogenic Ryleigh-Bénrd convection experiments [1, ]. Two of the most drmtic discoveries were the observtion of time dependence lmost immeditely bove the onset of convective flow, nd the power-lw fll-off in frequency for the power spectrl density derived from time series of globl mesurement of the temperture difference cross the fluid t fixed het flow [3 5]. However, these nd other importnt observtions remin poorly understood. The power-lw behvior ws unexpected, since bounded deterministic models typiclly show n exponentil flloff t high frequency [6]. Phenomenologicl stochstic models were proposed to explin the spectr [7], but no understnding of the origin of the d hoc stochstic driving hs followed. In this pper, we use numericl simultions of the three-dimensionl Boussinesq equtions for the fluid flow nd het trnsport in the cylindricl geometries of the experiments with relistic boundry conditions to investigte the power spectrum in more detil. The numericl simultions llow us to determine the sptil structure of the flow field in the periodic dynmics, nd the bsence of experimentl or mesurement noise provides us with Electronic ddress: mpul@cltech.edu more complete results for the power spectr. Our completely deterministic simultions yield results consistent with the experimentl observtions, including powerlw flloff of the power spectrum over the rnge ccessible to the experiment. Using knowledge of the flow field, we re ble to ssocite this power-lw behvior with specific events in the dynmics, nmely, the cretion nd nnihiltion of defects in the convection roll structure, which occur on time scle rpid compred with the slow pttern evolution. At higher frequencies, the power spectr decy exponentilly, consistent with the behvior expected for smooth deterministic time evolution. The low mplitude region of the spectr ws inccessible experimentlly due to the noise floor. Our simultions in cylindricl geometry re performed using n efficient spectrl element lgorithm (described in detil elsewhere [8]). The velocity u, temperture T, nd pressure p, evolve ccording to the Boussinesq equtions, ( ) σ 1 t + u u = p + RT ẑ + u, ) ( t + u T = T, u =0, where t indictes time differentition, ẑ is unit vector in the verticl direction, σ is the Prndtl number, nd R is the Ryleigh number. The equtions re nondimensionlized in the stndrd mnner using the lyer depth

2 h, the verticl diffusion time for het τ v, nd the constnt temperture difference cross the lyer T, s the length, time, nd temperture scles, respectively. All vribles in the following discussion re nondimensionl using this scling. The lower nd upper surfces (z =0, 1) re noslip nd re held t constnt temperture. The sidewlls re no-slip nd perfectly conducting [15], nd the initil conditions re smll rndom therml perturbtions of mgnitude 0. imposed upon n otherwise quiescent lyer, u =0,T =0. In nerly ll cryogenic experiments, for resons of incresed experimentl resolution, the het flux cross the convection lyer, Q, nd the temperture of either the upper or lower surfce re held constnt while mesurements of T (t) re mde. These mesurements re reported s R(t)/R c or T (t)/ T c, where T c is the temperture difference cross the lyer nd R c is the Ryleigh number t the convective threshold. Theoreticl clcultions, on the other hnd, most often consider both the upper nd lower surfces to be held t constnt temperture nd observe the time dependence in Q(t), which cn be reported s time series of the usselt number (t) (the normlized het current through the fluid lyer). It hs been shown experimentlly tht fixing Q or fixing T does not pper to chnge the flow dynmics, nd the conclusions from mesurements of R(t) t fixed or (t) t fixed R will be similr [9]. In order to mke contct with experiment [5, 9, 10] we focus our discussion on simultions with spect rtio Γ = 4.7 (Γ = r/h, r is the rdius), σ =0.78 (experimentl fluid ws nonsuperfluid He 4 ), nd constnt T. A key result of the experiments ws the observtion tht the power spectrum, P (), of mesured R(t) vlues exhibited the power-lw behvior, P () n,n = 4.0 ± 0. over the frequency rnge [10] (results were reported for ɛ = 3.6, where ɛ = (R R c )/R c is the reduced Ryleigh number). Six representtive time series (t) from our simultions re shown in Fig. 1. In terms of the horizontl diffusion time for het τ h (τ h =Γ τ v ), the simultion times re t f 100τ h (t f 50τ h for cse (vi)), which is comprble with the longest experiments, t f 65τ h (with one long run for t f 135τ h ) [5]. This is considerbly longer thn Γτ h, which hs been suggested s the erliest time scle for the flowfield to rech equilibrium [11]. However, s discussed below, we find tht the dynmics cn occur on even longer time scles. The simulted time-verged vlues of 1 re within 5.5% of the experimentl vlues given by 1=1.034β β β 5, β =1 R c /R [3]. Sptil nd temporl resolution studies hve been performed to ensure the ccurcy of the clculted vlues of (t) for the chosen simultion prmeters. We now consider the periodic time series in more detil, cse (ii) in Fig. 1. To determine the influence of the pttern dynmics on the power spectrum, we used sliding window in time to clculte successive time-loclized power spectr ( spectrogrm) s shown in Fig. b. Most vi t FIG. 1: Plots of the dimensionless het trnsport (t) for cses (i-vi) for reduced Ryleigh number ɛ = 0.557, 0.614, 0.8, 1.0, 1.5, nd 3.0, respectively. For cses (i-v), t =0.01, nd for cse (vi), t =0.005 ( t is the time step). of the power in the spectr cn be ttributed to nucletion of disloction pirs (nd, to lesser extent disloction nnihiltion). A plot of P () t prticulr time ( verticl slice of Fig. b) yields the windowed power spectrum centered bout tht point in time in Fig.. Figure 3 shows three such power spectr from the spectrogrm evluted with windows centered on (), (d), nd (e) corresponding to disloction nucletion, nnihiltion, nd glide, respectively. The locl power spectrum centered on the nucletion of disloction pir genertes power-lw region of significnt mgnitude; the locl power spectrum centered on the disloction nnihiltion genertes power-lw region tht is fctor of 10 smller in mgnitude; wheres the locl power spectrum clculted during disloction glide flls off more rpidly with nd does not mke significnt contribution. The origin of the power-lw is the time signture of the nucletion of disloction pir mnifested in the qusi-discontinuous slope of during the rpid excursion to stte of decresed het trnsfer (n individul event is shown in Fig. 4). ote tht tringulr feture with discontinuous chnges in the slope yields 4 symptotic behvior in the power spectrum. The power spectr of the remining time series in Fig. 1 yield power-lw region P () 4 becuse the dynmics (periodic, qusiperiodic, nd chotic) re dominted by the nucletion of disloction pirs nd roll pinch-off events. The differences between the chotic nd periodic dynmics re pprent in the low-frequency region nd re more pprent on log-liner plot of P (). For exmple, Fig. 4 shows the similrity between (t) for the periodic simultion, cse (ii), nd the chotic simultion, cse (vi). The dynmics in the chotic stte re i v iv iii ii

3 3 c d e b t FIG. : Time series (t) (top) nd corresponding spectrogrm (bottom) for one period of cse (ii). The lbels -f represent prticulr moments in the evolution of the pttern nd re discussed in the text. The spectrogrm displys 9 orders of mgnitude of the power, P (), with the smllest nd lrgest contours lbeled; the remining contours ech differ by fctor of 10. The spectrogrm ws clculted using sliding Hnn window of width t = 0.48 nd linerly detrended overlpping segments (segments overlp by t = 0.0). much more complicted; however, they re dominted by the roll pinch-off events tht mintin the chrcteristic qusi-discontinuous slope of, yielding power-lw region in the power spectrum. A comprison of the power spectr for the periodic nd chotic time series is shown in Fig. 3. The verge of the windowed power spectr of Fig. 3 eventully exhibit n exponentil decy, which continues until reching the spectrum floor; this is shown for cse (ii) in Fig. 5. Exponentil decy in the power spectr t high frequency is expected for bounded smooth deterministic dynmics [6]. The exponentil decy in the power spectr ws not detected in experiment due to the presence of instrumentl noise which msked the smll scle region. In the cryogenic experiments, flow visuliztion ws not possible leving the precise detils of the underlying pttern uncertin. With this in mind, we briefly discuss the dynmics represented in Fig. 1. Cse (i) illustrtes time-independent Pn-Am pttern similr to pnel () of Fig. 6. Cse (ii) is periodic with period t =8.4τ h (note the initil trnsient lsting 7τ h ); the dynmics of one period re illustrted in Figs. nd 6. Figure 6 displys the pttern t six different instnces in time corresponding to the events lbeled in Fig.. Initilly there is Pn-Am pttern with two opposing wll foci cusing roll compression (), eventully nucleting disloction pir in the center of the domin (b). The disloctions quickly climb to the wll (c), t which point they both begin f P() e (ii) d FIG. 3: Windowed power spectr. The power spectr lbeled (), (d), nd (e) re verticl slices of the spectrogrm tken t representtive times for cse (ii) t disloction nucletion, disloction nnihiltion, nd disloction glide, respectively. The curves lbeled (ii) nd (vi) re the verge of the windowed power spectr using the entire spectrogrm for cses (ii) nd (vi). The dshed lines represent P () 4. to glide slowly towrd the sme wll focus. However, the lower disloction is nnihilted t the sidewll (d), nd the remining disloction continues to glide slowly into the wll focus, where it is nnihilted (e). A Pn- Am pttern gin forms (f), nd finlly the process repets. This is in generl greement with flow visuliztions from relted room-temperture rgon-bsed experiments [1, 13]. Cse (iii) my be periodic on long time scle of t 40τ h ; the durtion of the simultion is indequte to be conclusive. Cse (iv) illustrtes chotic burst of durtion t 54τ h bounded by periodic dynmics with period of t 17. Agin the simultion durtion is indequte to determine whether this is trnsient stte or whether the chotic bursting will repet. Cse (v) shows n initil chotic trnsient tht mkes trnsition t t 18τ h to very complicted qusiperiodic stte where the centrl roll pir is pinned by the dynmic motion of two opposing disclintions. The dominnt mode in the qusiperiodic stte hs time scle t 8. Cse (vi) illustrtes chotic dynmics. We hve lso performed simultions for the Γ = 4.7 cylindricl domin with insulting lterl boundry conditions, in ddition to simultions in Γ = 7.66 domin (σ = 0.69 for rgon) for both conducting nd insulting lterl boundries. Considering these dditionl results, we mintin our conclusions concerning the origin of the power-lw. This work represents joint computtionl nd theoreticl effort to further our quntittive understnding of complex dynmics in sptilly extended nonequlibrium systems. An importnt link missing in nerly ll theoreticl work to dte hs been quntittive comprison with experiment. Our results demonstrte tht this quntittive comprison with experiment is now possible. We pln to use this pproch to investigte sptiotemporl chos in lrger spect rtio systems. (vi)

4 b t * c d FIG. 4: A closeup of the time series (t), illustrting the signture of nucletion of disloction pir for cse (ii) (solid line), nd the signture of roll pinchoff for cse (vi) (dshed line). Time t is mesured s t =(t t i )/ t; t i denotes when the event begins, nd t is the durtion of the event. For the events shown, t i = 587, 70.5 nd t =18, 1.5 for cses (ii) nd (vi), respectively. e f This reserch ws supported by the U.S. Deprtment of Energy, Grnt DE-FT0-98ER1489, nd the Mthemticl, Informtion, nd Computtionl Sciences Division subprogrm of the Office of Advnced Scientific Computing Reserch, U.S. Deprtment of Energy, under Contrct W Eng-38. We lso cknowledge the Cltech Center for Advnced Computing Reserch nd the orth Crolin Supercomputing Center. FIG. 6: Flow visuliztion showing contours of the therml perturbtion t the mid-depth, (6 evenly spced contours; 0. δt 0., negtive vlues re dshed lines, nd positive vlues re solid lines) for cse (ii). Pnels -f re for t = 600, 605, 630, 650, 735, 785. The disloctions glide to the right; during the next period, the disloctions glide to the left, s cn lredy be discerned in (f) by the bis in the roll compression. This left nd right lterntion continues for the entire simultion. P() FIG. 5: The power spectrum, P (), for cse (ii) on log-liner scle to illustrte region of exponentil decy. The slope of the dshed line is -3.8; the crossover to exponentil decy occurs t 1.5. REFERECES [1] M. C. Cross nd P. C. Hohenberg, Rev. of Mod. Phys. 65, 851 (1993). [] R. P. Behringer, Rev. Mod. Phys. 57, 657 (1985). [3] G. Ahlers, Phys. Rev. Lett. 33, 1185 (1974). [4] G. Ahlers nd R. P. Behringer, Phys. Rev. Lett. 40, 71 (1978). [5] G. Ahlers nd R. W. Wlden, Phys. Rev. Lett. 44, 445 (1980). [6] U. Frisch nd R. Morf, Phys. Rev. A 3, 673 (1981). [7] H. S. Greenside, G. Ahlers, P. C. Hohenberg, nd R. W. Wlden, Physic, D. 5, 3 (198). [8] P. F. Fischer, J. Comp. Phys. 133, 84 (1997). [9] H. Go nd R. P. Behringer, Phys. Rev. A 30, 837 (1984). [10] G. Ahlers nd R. P. Behringer, Prog. Theor. Phys. Supp. 64, 186 (1978). [11] M. C. Cross nd A. C. ewell, Physic D 10, 99 (1984).

5 5 [1] V. Croquette, P. Le Gl, nd A. Pocheu, Phys. Scr. T13, 135 (1986). [13] A. Pocheu, J. Phys. Frnce 50, 059 (1989). [14] R. P. Behringer nd G. Ahlers, J. Fluid Mech. 15, 19 (198). [] The cryogenic experiments [14] were bounded by thin stinless steel lterl wll seprting the convective lyer from surrounding vcuum resulting in the diversion of 0% of the het flow suggesting n insulting boundry. The room-temperture rgon experiments [1, 13], on the other hnd, were bounded by highly conducting sidewlls. We hve performed simultions using both lterl boundry conditions nd find tht the generl nture of the resulting dynmics is unffected.

NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.

NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by. NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with