ORE Open Reearch Exeter TITLE The onet of convection in rotating circular cylinder with experimental boundary condition AUTHORS Zhang, Keke; Liao, X. JOURNAL Journal of Fluid Mechanic DEPOSITED IN ORE 4 March 213 Thi verion available at http://hdl.handle.net/136/4397 COPYRIGHT AND REUSE Open Reearch Exeter make thi work available in accordance with publiher policie. A NOTE ON VERSIONS The verion preented here may differ from the publihed verion. If citing, you are advied to conult the publihed verion for pagination, volume/iue and date of publication
J. Fluid Mech. (29), vol. 622, pp. 63 73. c 29 Cambridge Univerity Pre doi:1.117/s221128517x Printed in the United Kingdom 63 The onet of convection in rotating circular cylinder with experimental boundary condition KEKE ZHANG 1 AND XINHAO LIAO 2 1 Department of Mathematical Science, Univerity of Exeter, Exeter EX4 4QE, UK 2 Shanghai Atronomical Obervatory, Chinee Academy of Science, Shanghai 23, China (Received 1 Augut 28 and in revied form 27 October 28) Convective intabilitie in a fluid-filled circular cylinder heated from below and rotating about it vertical axi are invetigated both analytically and numerically under experimental boundary condition. It i found that there exit two different form of convective intabilitie: convection-driven inertial wave for mall and moderate Prandtl number and wall-localized travelling wave for large Prandtl number. Aymptotic olution for both form of convection are derived and numerical imulation for the ame problem are alo performed, howing a atifactory quantitative agreement between the aymptotic and numerical analye. 1. Introduction Motivated by the wih to undertand the fundamental dynamic taking place in planetary fluid interior and atmophere, convection in rotating cylindrical geometry ha been extenively tudied both experimentally and theoretically (ee, for example, Daniel 198; Zhong, Ecke & Steinberg 1991; Herrmann & Bue 1993; Kuo & Cro 1993; Aurnou et al. 23; Plaut 23; Young & Read 28). In particular, careful numerical tudie of convective intabilitie in a fluid contained in a circular cylinder uniformly heated from below and rotating about it vertical axi (Goldtein et al. 1993, 1994) reveal an exceedingly complicated and eemingly incomprehenible behaviour at mall and moderate Prandtl number. We have recently undertaken the tudy of convective intabilitie in a rotating circular cylinder with tre-free end (Zhang, Liao & Bue 27). The aumption of the tre-free boundary condition permit eparable olution and, conequently, lead to a much implified mathematical problem governed by ordinary differential equation. It reveal that inertial convection i preferred for fluid with mall and moderate Prandtl number if the tre-free condition i aumed. Since inertial convection i energetically driven by thermal buoyancy againt vicou diipation that occur largely in the vicou boundary layer (ee, for example, Zhang 1995; Bue & Simitev 24), the type of velocity boundary condition play, both phyically and mathematically, an eential role in determining the main propertie of convection. The Ekman boundary layer reulting from the non-lip top and bottom not only introduce trong ma fluxe into the interior but alo bring dominant term in the aymptotic Email addre for correpondence: kzhang@ex.ac.uk
64 K. Zhang and X. Liao olution of inertial convection in rapidly rotating cylinder. It i therefore deirable to gain a better undertanding of convection under experimental boundary condition, the problem then being governed by partial differential equation, providing a ueful theoretical framework and guidance for experimental tudie of the problem. Thu the primary objective of thi tudy i to undertand, through both numerical and aymptotic analyi, convective intabilitie in a rotating cylinder under experimental boundary condition. Our focu will be mainly on intabilitie at mall and moderate Prandtl number. However, the aymptotic olution for large Prandtl number i alo briefly dicued for the purpoe of completene. It i fortunate that on recognizing and utilizing the phyical nature of convective intabilitie, we are able to derive everal relatively imple aymptotic formula that correctly decribe the key feature of convection for all value of the Prandtl number and the apect ratio and which agree atifactorily with the reult of numerical analyi. In what follow we hall begin by preenting the mathematical equation in 2. The numerical analyi i briefly preented in 3, while the correponding aymptotic analyi i dicued in 4, with a ummary and everal remark given in 5. 2. Mathematical formulation Conideration i given to convective intabilitie in a Bouineq fluid confined in a circular cylinder of radiu o and height d with contant thermal diffuivity κ, thermal expanion coefficient α and kinematic vicoity ν. The circular cylinder rotate uniformly about it axi with a contant vertical angular velocity Ω in the preence of a contant gravitational field g = gẑ, and i heated uniformly from below to produce an untable vertical temperature gradient, T = β ẑ, where β i a poitive contant. Cylindrical coordinate (, φ, z) with the correponding unit vector (ŝ, ˆφ, ẑ, whereẑ i parallel to the axi of rotation) are ued. Making ue of the depth d a the length cale, Ω 1 a the unit of time and βd 3 Ω/κ a the unit of temperature fluctuation, the problem of linear convective intability i governed by the equation u t +2ẑ u = p + RΘẑ + E 2 u, (2.1) u =, (2.2) Pr Θ = E(ẑ u + 2 Θ), (2.3) t where u i the three-dimenional velocity field, and the temperature deviation from the purely conductive tate T (z) i repreented by Θ. In cylindrical coordinate, we write olution of the equation in the form (u,θ)(, φ, z, t)= (u,θ) (, z) exp [i(2σt + mφ), where m i the azimuthal wavenumber aumed to be poitive and σ i the half-frequency which i ued becaue of it property σ < 1 in the limit E =. The three non-dimenional phyical parameter the Rayleigh number R, the Prandtl number Pr andtheekmannumbere are defined a R = αβgd2 Ωκ, Pr = ν κ, E = ν Ωd. 2 The geometric parameter i given by the apect ratio, defined a = /d. Appropriately for experiment of the problem (ee, for example, Goldtein et al. 1993), we hall aume no-lip velocity and perfectly conducting condition on the
Convection in rotating circular cylinder 65 top and bottom of the cylinder, u = Θ = at z =, 1. (2.4) On the idewall, we aume no-lip and perfectly inulating condition which require u = Θ = at =. (2.5) Equation (2.1) (2.3) ubject to the boundary condition in (2.4) (2.5) form the convective intability problem which will be olved by both aymptotic and numerical analyi at, a in practical computation and phyical experiment, a mall but fixed E for different value of Pr. 3. Numerical analyi Compared to the numerical analyi for tre-free end, which allow olution in eparable variable, the numerical analyi for the no-lip condition require olution of partial differential equation and hence i much more complicated. In order to determine the mot untable mode of convective intabilitie, we mut conider diturbance of all poible azimuthal wavenumber. Axiymmetric convection may be phyically preferred in rotating pherical ytem (ee, for example, Net, Carcia &Sánchez 28). However, our extenive numerical and aymptotic tudie how that axiymmetric (m =) intabilitie are not preferred for any value of Pr when E 1and = O(1). In conequence, we hall preent the numerical and aymptotic analyi only for non-axiymmetric (m ) olution. For non-axiymmetric intabilitie with m, the velocity vector u atifying (2.2) can be expreed in term of two calar potential Ψ and Φ (Marqué 199): u = 1 ( Ψ Φ φ ŝ + z Ψ ) ˆφ 1 Φ ẑ. (3.1) φ In term of Ψ and Φ, the non-lip condition on the idewall and the end become Ψ = Ψ = Φ = at = ; Ψ = Φ z = Φ = at z =, 1. (3.2) Making ue of the expreion (3.1) and applying ẑ and ŝ onto (2.1), we can derive the three independent non-dimenional calar equation: ( )[ ( 1 t E 2 Φ ) ) ( 2 2 Ψ + 2 2 Φ =, (3.3) z z 2 z φ [ ( t E 2 + 2 2 2 Ψ z φ 2E + 1 ) [ 2 Ψ 2 z [ ( ) 1 2 Φ z 2 [ E 2 Pr t ( 2 1 ( 2 2 Θ E Φ φ z 2 ) Ψ z ) Φ R Θ φ =, (3.4) =. (3.5) The above equation are then olved numerically by uing the Galerkin-type method in which, for example, Ψ and Φ are expanded in term of the Chebyhev function
66 K. Zhang and X. Liao R c 1 2 1 1 (1. 1 6, 3,1) (1. 1 4, 3,1) (1. 1 3, 5,1) (2.5 1 3, 8,1) (5. 1 3, 7,2) (1. 1 2, 5,4) (2.5 1 2, 4,6) (6.8 1 2, 3,8) 1 1 6 1 4 1 2 1 Pr Figure 1. The critical Rayleigh number R c i plotted a a function of Pr in a rotating cylinder with =2andE =1 4. Solid line repreent the numerical olution while dahed line reult from the aymptotic olution. Of the three number in the bracket, the firt i Pr, the econd i m and the third i the number of node in the direction. T j (x): [ J Ψ = m (1 š) 2 (1 ž 2 ) j= L Ψ mj l T j (ž) T l (š) e i(mφ+2σt), (3.6) l= [ J L Φ = m+1 (1 š)(1 ž 2 ) 2 Φ mj l T j (ž) T l (š) e i(mφ+2σt), (3.7) j= l= where ž =2z 1, š =2/ 1andJ = L = O(1) for achieving an accuracy within 1 %, the factor m and m+1 are impoed for regularity of the olution at = and the complex coefficient like Ψ mj l are obtained, together with the Rayleigh number R and the half-frequency σ, by a tandard numerical procedure. The primary aim of our numerical analyi i to provide a valuable comparion with the reult of the aymptotic analyi valid only for mall E. Some typical reult of the numerical analyi, along with the aymptotic reult, are preented in figure 1, howing the critical Rayleigh number R c for the mot untable mode of convective intabilitie a a function of Pr in a rotating cylinder with =2 and E =1 4. The convection pattern for three typical Prandtl number, Pr=.25,.25 and 1., are depicted in figure 2. We hall dicu the relevant detail when comparing them with the aymptotic olution. 4. Aymptotic analyi A key aumption in the aymptotic analyi i that the velocity of convection at leading order for mall or moderate Pr can be repreented by a ingle inertial-wave mode while buoyancy force appear only at the next order to drive the inertial wave againt the effect of vicou damping, leading to an aymptotic expanion in the form } u = u + u b + u 1, p= p + p b + p 1, (4.1) σ = σ + σ 1, Θ= Θ +, R = R 1 +, where u 1, p 1 and σ 1 repreent mall interior vicou correction to the leading-order invicid interior olution u, p and σ, while u b and p b denote olution of the vicou boundary layer which are non-zero only in the vicinity of the bounding urface of
Convection in rotating circular cylinder 67 (a) (c) (e) (b) 1..8.6 z.4.2.2.4.6.8 1. 1.2 1.4 1.6 1.8 2. (d) 1..8.6.4.2.2.4.6.8 1. 1.2 1.4 1.6 1.8 2. ( f ) 1..8.6.4.2.2.4.6.8 1. 1.2 1.4 1.6 1.8 2. Figure 2. Contour of u φ in a horizontal plane at z =.75 (upper panel) and in a vertical plane (lower panel): (a,b) forpr =.25, (c,d) forpr =.25 and (e,f )forpr =1. with =2 and E =1 4. the cylinder. More preciely, u = O(1), u b = O(1) and u 1 1, with the flux from the ocillatory boundary layer u b providing the matching condition for the econdary interior flow u 1.ItiofimportancetonotethatE 1/2 cannot be ued a an expanion parameter in thi problem becaue the thickne of an ocillatory vicou boundary layer i not generally of order E 1/2, the flux from the ocillatory boundary layer i not generally of order E 1/2 and the internal vicou contribution i not generally of the order E 1/2 maller than that of the ocillatory boundary layer (Zhang & Liao 28). In other word, the effect of patial and temporal non-uniformitie obcure the form of the aymptotic expanion even for the firt vicou corrective term. After ubtitution of the expanion into (2.1) (2.3), the leading-order problem, ubject to the uual invicid boundary condition, i given by u +2ẑ u + p =, (4.2) t u =, (4.3) Pr Θ = E(ẑ u + 2 Θ ), (4.4) t which decribe non-diipative thermal inertial wave with olution in the form [ i ŝ u = 2 ( σ ξ ) 1 σ 2 [ 1 ξ ˆφ u = 2 ( ) 1 σ 2 J m 1( )+ m(1 σ ) J m ( ) J m 1( ) m(1 σ ) J m ( ) co(πz) e i(mφ+2σt), (4.5) co(πz) e i(mφ+2σ t), (4.6) ẑ u = iπ J m ( ) in(πz) e i(mφ+2σt), 2σ (4.7) N 2im ( ξ 2 βmn) 2 Qmn J m (ξ) in(πz) J m (β mn / ) Θ = e i(mφ+2σt), π 2 [π 2 +(β mn / ) 2 +2PrE 1 σ i J m (β mn ) (4.8) n=1 p = J m ( )co(πz) e i(mφ+2σ t), (4.9)
68 K. Zhang and X. Liao where = ξ/ (the implet vertical tructure that i relevant to convective intabilitie i taken), J m (x) denote the tandard Beel function, β mn are the root of J m(β mn ) = with <β m1 <β m2 <β m3, Q mn =(πβ mn ) 2 /[2σ 2(ξ 2 βmn) 2 2 (βmn 2 m 2 ), N O(1) for achieving an accuracy within 1 % and ξ i a olution of [ ξj m 1 (ξ)+ σ ( ) 2 1/2 ξ 1+ 1 σ π mj m(ξ) =, (4.1) σ = ±[1 + (ξ/π) 2 1/2. (4.11) Different value of ξ or σ correpond to different inertial mode. At the next order, the equation governing the perturbation of the interior flow are i2σ u 1 +2ẑ u 1 + p 1 = R 1 ẑθ + E 2 u i2σ 1 u, (4.12) u 1 =. (4.13) It i traightforward to derive the olvability condition required for the inhomogeneou equation (4.12): [ E 1/2 p ˆn ( ˆn u b )dη ds = S V [ R1 u ẑθ 2iσ 1 u 2 E u 2 dv, (4.14) where u and p denote the complex conjugate of u and p, V volume integral over the cylinder, repreent a urface integral over it bounding S urface with the normal vector ˆn and η i the tretched boundary-layer coordinate. In deriving (4.14), the ma flux from the Ekman boundary layer i ued to provide the matching condition for u 1 at the outer edge of the layer. Each integral in (4.14) can be evaluated. Upon uing (4.5) (4.8) together with the propertie of the Beel function, we can how that N [ 2 u πq mn mjm (ξ) ẑθ dv =, π 2 +(β mn / ) 2 +2iPrE 1 σ σ (4.15) V V n=1 u 2 dv = σ 2 π 2 V u 2 dv = π[(π )2 + m(m σ )J ( ) m(ξ) 2. (4.16) 4σ 2 1 σ 2 In order to evaluate the urface integral on the left-hand ide of (4.14), we mut derive the boundary-layer flow u b on the bounding urface of a cylinder. Conider firt the Ekman boundary layer on the bottom urface at z = decribed by ) ) (i2σ 2 ẑ u η 2 b =2u b, (i2σ 2 u η 2 b = 2ẑ u b, (4.17) where η = E 1/2 z and u b = u at z = o that the no-lip condition i obeyed. Note that η = i at the bottom urface while η = i at the outer edge of the boundary layer but till located at the bottom in term of the coordinate z. The boundary-layer olution to (4.17) on the bottom urface can be expreed in the form {[ 1 u b = 4 ( ξ(σ 1) ) J 1 σ 2 m 1 ( )+ 2m(1 σ ) J m ( ) (iŝ + ˆφ)e (1+i)η 1+σ [ } ξ(σ +1) + J m 1 ( ) (iŝ ˆφ)e (1 i)η 1 σ e i(mφ+2σt), (4.18)
Convection in rotating circular cylinder 69 by which, together with (4.9), we can obtain the contribution from the vicou boundary layer at z = to the integral on the left-hand ide of (4.14): 2π [ [ (p) πj 2 z= ˆn ( ˆn u b )dη d dφ = ( m(ξ) ) 1 σ 2 8σ 2 {( 1+σ + 1 σ )C 1 2σ ( 1+σ 1 σ )C 2 +i[( 1+σ 1 σ )C 1 2σ ( 1+σ + 1 σ )C 2 }, (4.19) where C 1 =(1+σ 2)(m2 + π 2 2 ) 2mσ and C 2 = π 2 2 + m(m σ ). Becaue of the ymmetry between the top and the bottom of the cylinder, the contribution from the vicou boundary layer at z = 1 i exactly the ame a that from the boundary layer at z =. Similarly, we can derive the contribution from the idewall boundary layer to the integral on the left-hand ide of (4.14): 2π 1 [ (p) = ˆn ( ˆn u b )dη dz dφ = π 4σ σ ( i+ σ ) (m 2 + 2 π 2 )J 2 σ m(ξ). (4.2) Subtitution of (4.15), (4.16), (4.19) and (4.2) into the olvability condition (4.14) yield a complex equation whoe real part determine R 1 while the imaginary part give rie to the vicou correction σ 1 : [ E 1/2 N ( π 2 + β R 1 = mn/ 2 2) 1 Q mn 4m 2 ( 1 σ 2 n=1 π2 + βmn/ 2 2) 2 +(2σ Pr/E) 2 { E 1/2 π 2 [(π ) 2 + m(m σ ) 2σ m[ 1+σ + 1 σ 1 σ 2 σ 2 [ } +(m 2 + π 2 2 σ (1 σ 2 ) ) +(1+σ ) 3/2 +(1 σ ) 3/2, (4.21) ( ) 1 σ 2 1/2E {[ 1/2 N 2σ =2σ Q mn π 2 2 ( + m(m σ ) π2 + βmn/ 2 2) 2 +(2σ Pr/E) 2 n=1 [8m 2 R PrE 3/2 σ ( 1 σ 2 ) 1/2 +2σ m[ 1+σ 1 σ +(m 2 + π 2 2 ) [ σ 1 σ 2 σ } (1 + σ ) 3/2 +(1 σ ) 3/2. (4.22) In addition to the expreion for the Rayleigh number R 1 and the frequency σ,the leading-order velocity of convection can be explicitly written a
7 K. Zhang and X. Liao { [ i ŝ u = 4 ( σ ξ ) 2 1 σ 2 J m 1 ( ) + m(1 σ ) J m ( ) co πz [ ξ(σ 1) + J m 1 ( ) + 2m(1 σ ) [e χ J m ( ) +z e χ + (1 z) + ξ(σ +1) J m 1 ( ) [ e χ z e } χ (1 z) e i(2σt+mφ), (4.23) { [ 1 ˆφ u = 4 ( ξjm 1 ( ) ) 2 m(1 σ ) J 1 σ 2 m ( ) co πz + mj m ( ) ) co πze χ( 2σ [ ξ(σ 1) + J m 1 ( ) + 2m(1 σ ) [e χ J m ( ) +z e χ + (1 z) ξ(σ +1) J m 1 ( ) [ e χ z e } χ (1 z) e i(2σt+mφ), (4.24) ẑ u = iπ 2σ [ Jm ( ) J m (ξ) e χ( ) in (πz) e i(2σt+mφ), (4.25) where χ + =(1 + i) E 1 (1 + σ ), χ =(1 i) E 1 (1 σ ) and χ = σ E 1 (1 + iσ / σ ). A complete analytical olution of the problem i decribed by (4.21) (4.25). A indicated by (4.21), the Rayleigh number R 1 required to initiate inertial convection increae rapidly with increaing Pr and the idewall-localized convection would become phyically preferred at large Pr. It i worth mentioning that the aymptotic analyi (Herrmann & Bue 1993) wa concerned with the leading-order olution for which the effect of the non-lip condition wa not conidered. Since cylindrical geometry play an inignificant role in the wall-localized convective flow, we can imply re-cale the aymptotic olution in a channel obtained by Liao, Zhang & Chang (26), giving rie to R =2π 6 2 3+73.8E 1/3, (4.26) σ =[π 2 3 2+ 3E/Pr 29.6E 4/3 Pr 1, (4.27) m = [ π 2+ 3 27.76E 1/3, (4.28) which ha taken into account the non-lip boundary condition while it leading order i the ame a that given by Herrmann & Bue (1993). For any given value of, Pr and E, the following three tep are uually required to find the mot untable mode of convective intabilitie: (a) ue (4.21) to calculate the mallet R 1 required for the non-axiymmetric inertial convection over different value of σ,i.e. over different inertial mode; (b) ue (4.26) to calculate the econd Rayleigh number required for the wall-localized convection and (c) compare the two Rayleigh number obtained at exactly the ame, Pr and E. Of phyical ignificance i the minimum value of the Rayleigh number, which i denoted by R c and will be referred to a the critical Rayleigh number, at which convective intabilitie firt et in when the Rayleigh number R gradually increae from zero. The R c a a function of Pr, calculated from both the aymptotic and the numerical analyi, i hown in figure 1 for E =1 4 and = 2: all the mot untable mode of convection are non-axiymmetric. The azimuthal wavenumber
Convection in rotating circular cylinder 71 m c and the correponding node in the radial direction for everal typical Prandtl number are alo hown in figure 1. For example, the aymptotic formula (4.21) (4.22) at Pr=.25 yield R c =3.41,m c =8,σ c =.474 with σ =.4773, while the numerical analyi give R c =3.58,m c =8,σ c =.473, with the patial tructure of it flow diplayed in figure 2(a,b). Note that there are no noticeable difference between the numerical olution and the correponding analytical olution (4.23) (4.25) when they are plotted. When Pr increae, the critical wavenumber m c may decreae at the expene of increaing the radial complexity of the flow. At Pr=.25, the aymptotic expreion (4.21) (4.22) give rie to R c =2.4,m c =4,σ c =.248 with σ =.2634 and ix node in the radial direction, whoe patial tructure i hown in figure 2(c,d). The correponding numerical olution i characterized by R c =21.6,m c =4,σ c =.247, howing nearly the identical patial tructure. When Pr increae to about.7 at E =1 4, a hown in figure 1, the aymptotic olution given by (4.21), which i till mathematically valid and give a atifactory approximation to the problem, ceae to be phyically relevant becaue the walllocalized convection given by (4.26) (4.28) become preferred. A typical wall-localized olution at Pr=1. fore =1 4 and = 2 i hown in figure 2(e,f ). In thi cae, the aymptotic analyi give R c =63.6,m c =9.56 and σ c =.2, while the numerical analyi yield R c =62.6,m c =1 and σ c =.24. Moreover, the agreement between the aymptotic and numerical analyi i expectedly better for maller value of E. For example, at E =1 5 with =2 and Pr=.1, the aymptotic expreion (4.21) (4.22) yield R c =15.,m c =2,σ c =.15 while the numerical analyi give R c =15.4,m c =2,σ c =.15. Our analyi reveal that there exit three different regime of convection in a rotating cylinder at a given mall E. The firt i the regime of inertial convection at very mall Pr, i.e. Pr/E O(1). A diplayed in the left portion of figure 1 for Pr 1 4, a well a indicated by the aymptotic expreion (4.21), the critical Rayleigh number R c i nearly independent of Pr with R c = O(E 1/2 ). In thi regime, the temperature Θ i marked by approximately the ame phae a that of the vertical flow u z in the convective heat tranport. But thi ituation i changed when Pr/E>O(1), which i hown in the middle portion of figure 1 for 1 4 <Pr<1 1. Although the inertial effect i till predominant, the temperature Θ i marked by a large phae hift comparing to the vertical flow u z and, conequently, the R c increae ignificantly with increaing Pr. In thi moderate-pr regime, the aymptotic expreion (4.21) indicate that there exit no imple aymptotic relation between R c and E or between R c and Pr. It i alo in thi regime that convection pattern i not only highly complex but alo trongly dependent on Pr. Thi complicated behaviour tem from the exitence of a large manifold of two-dimenional inertial mode that may be excited by convective intabilitie in thi regime, indicating how difficult any numerical attempt to pinpoint it coherent tructure will be. For larger value of Pr, which i hown in the right portion of figure 1 for Pr>1 1, the trong vicou effect reulting from the walllocalized boundary-layer flow i dynamically dominant, leading to the relatively imple aymptotic dependence. Our calculation for different value of E ugget that thi picture i qualitatively correct for = O(1) at any given mall Ekman number E. 5. Summary and remark The problem of convection in a rotating circular cylinder heated from below and rotating about a vertical axi ha been tudied a a paradigm for undertanding the general dynamic of rotating convection. We have undertaken both numerical and
72 K. Zhang and X. Liao analytical tudie of convective intabilitie under experimental boundary condition, howing that the intabilitie are either of inertial-wave type or of wall-localized type. In the aymptotic tudy, the primary aumption i that the convective flow at mall or moderate Pr can be repreented by an inertial-wave mode. In the numerical tudy, we have olved the partial differential equation governing the convection problem. Agreement between the analytical and numerical olution ha been hown to be atifactory. There are phyical and mathematical conequence of our aymptotic analyi for convection in rotating cylinder. Phyically peaking, when a cylinder rotate rapidly (E 1), the fundamental dynamic of fluid motion can be intuitively illutrated by the Proudman Taylor theorem tating that infiniteimal teady motion in rotating invicid fluid are two-dimenional with repect to the direction of ẑ. It follow that u z / z = and that, upon applying the boundary condition on the top and bottom of a cylinder, we mut conclude that u z = within the cylinder. In other word, convection cannot take place: the effect of rotation trongly contrain and tabilize the ytem. The contraint mut be broken o that convection can occur. Our aymptotic analyi how that it can be broken either by inertial effect for fluid with mall Prandtl number or by vicou effect for fluid with large Prandtl number in connection with the boundary-layer-type convection. Mathematically peaking, the baic aumption that the convection velocity at leading order for mall or moderate Pr can be repreented by an inertial-wave mode implie that the preent tability analyi i quite different from the uual one uch a that for the Rayleigh Bénard problem. In our tability analyi, it i not a quetion of imply finding a wavenumber m that minimize the Rayleigh number and then determine the mot untable convection mode. It i a quetion of finding the two-dimenional tructure of a leading-order flow that give rie to the minimum Rayleigh number. Thi explain why it i o difficult to elucidate the parametric dependence of the problem numerically. Moreover, ince the aymptotic formula can be readily evaluated, they would provide a helpful guidance for the experimental tudie of convection in rotating circular cylinder. Thi tudy repreent the firt aymptotic analyi for convective intabilitie in a rotating cylinder with experimental boundary condition for mall and moderate Prandtl number. It i ignificant to note that the two et of aymptotic formula given in 4 cover the whole parameter regime of the problem: all value of the apect ratio and the Prandtl number Pr at any fixed mall Ekman number E. Epecially, it i omewhat urpriing that the eemingly incomprehenible and highly complicated convection revealed numerically, for example, by Goldtein et al. (1993, 1994) can be atifactorily decribed by everal relatively imple analytical formula. Keke Zhang i upported by UK PPARC/NERC grant and Xinhao Liao i upported by NSFC/16333, STCSM/8XD1452 and CAS grant. The numerical computation i upported by SSC. We would like to thank Profeor F. H. Bue for hi helpful dicuion. REFERENCES Aurnou,J.M.,Andreadi,S.,Zhu,L.&OlonP.L.23. Experiment on convection in Earth core tangent cylinder. Earth Planet. Sci. Lett. 212, 119 134. Bue,F.H.&SimitevR.24. Inertial convection in rotating fluid phere. J. Fluid Mech. 498, 23 3. Daniel, P. G. 198. The effect of centrifugal acceleration on axiymmetric convection in a hallow rotating cylinder or annulu. J. Fluid Mech. 99, 65 84.
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