Viscous symmetric stability of circular flows

Save this PDF as:

Size: px
Start display at page:

Download "Viscous symmetric stability of circular flows"


1 J. Fluid Mech. (1), vol. 65, pp c Cambidge Univesity Pess 1 doi:1.117/s iscous symmetic stability of cicula flows R. C. KLOOSTERZIEL School of Ocean and Eath Science and Technology, Univesity of Hawaii, Honolulu, HI 968, USA (Received July 9; evised 18 Decembe 9; accepted Decembe 9) The linea stability popeties of viscous cicula flows in a otating envionment ae studied with espect to symmetic petubations. Though the use of an effective enegy o Lyapunov functional, we deive sufficient conditions fo Lyapunov stability with espect to such petubations. Fo cicula flows with swil velocity () we find that Lyapunov stability is detemined by the popeties of the function F() = (/ + f )/Q (with f the Coiolis paamete, the adius and Q the absolute voticity) instead of the customay Rayleigh disciminant Φ() =(/ + f )Q. The conditions fo stability ae valid fo flows with non-zeo Q eveywhee. Futhe, the flows ae pesumed stationay, incompessible and velocity petubations ae equied to vanish at igid boundaies. Fo Lyapunov stable flows an uppe bound fo the incease of the total petubation enegy due to tansient non-modal gowth is deived which is valid fo any Reynolds numbe. The theoy is applied to Couette flow and the Lamb Oseen votex. 1. Intoduction In Kloosteziel & Canevale (7) we studied the linea stability of some simple baoclinic paallel shea flows in otating statified systems though the use of appopiate Lyapunov functionals. The flows wee assumed zonally invaiant and subjected to petubations that ae also zonally invaiant. The instability that may o may not ensue is sometimes called inetial instability. It is closely elated to centifugal instability, which is a well-known obust phenomenon obseved in cicula flows subjected to petubations that ae also ciculaly symmetic. In the laboatoy centifugal instability manifests itself, fo example, as the famous axisymmetic Taylo votices in Couette flow, i.e. the flow between two co-axial otating cylindes. Because of the zonal invaiance of distubances in paallel shea flows and the ciculaly symmetic natue of the distubances of cicula flows, both instabilities ae appopiately called symmetic instability. In the olde meteoological liteatue it is sometimes called dynamic instability. Fo gaphs fom numeical simulations showing the tooidal ovetuning motions associated with the centifugal instability in votices and thei fully nonlinea evolution (see e.g. Kloosteziel, Canevale & Olandi 7). A sketch of the ovetuning motions associated with the tooidal votices is povided in figue 1(a). Unlike in Kloosteziel & Canevale (7), in this pape we conside the moe complicated question of stability of abitay cicula flows but we estict ouselves to flows in a homogeneous fluid. addess fo coespondence:

2 17 R. C. Kloosteziel (a) f (b) e z z v w θ u e e θ Unstable egion Figue 1. (a) Symmetic instability stats in an unstable egion as tooidal votices ( ib votices ) of altenating sign. Fo a baotopic votex with swil velocity () the unstable egion is whee the Rayleigh disciminant Φ() < with Φ() defined in (1.1). (b) Diagam defining the pola coodinate system and velocity components fo a cicula votex. Hee v is the swil velocity, w the vetical velocity component and u the adial velocity component. Fo abitay baotopic cicula flows with swil velocity (), the inviscid classical condition fo stability is that fo all : Φ() >, whee ( ) Φ = + f Q and Q = d d + + f = ω + f (1.1) is the absolute voticity. The inviscid classical condition fo instability is that Φ() < in some egion. The cylindical pola coodinate system (, θ, z) is sketched in figue 1(b). The elative voticity component ω is in the z diection and aligned with the axis of otation and gavitational acceleation. At the time unfamilia with the olde meteoological liteatue, Kloosteziel & van Heijst (1991) ediscoveed the citeion in an attempt to explain the difficulty in ceating stable anticyclones in a otating fluid in the laboatoy. When f = the condition Φ() < educes to Rayleigh s celebated ciculation citeion (Rayleigh 1916) fo symmetic instability (see Dazin & Reid 1981). In an extension of Bayly s analysis (Bayly 1988) to otating systems, Sipp & Jacquin () genealized the citeion to non-cicula flows. In Φ the tem / is eplaced by /R, with the velocity amplitude along a steamline and R the local algebaic adius of cuvatue of the steamline. If thee ae steamlines along which Φ<, then locally instability is guaanteed. The classical citeion (1.1) fo baotopic cicula flows in a homogeneous fluid follows fom the classical condition fo baoclinic cicula flows with swil velocity (, z) in a stably statified fluid. If the fluid is Boussinesq, the citeion fo stability is that eveywhee in the domain: N + Φ> and N Φ S >, whee S = ( ) + f z, (1.) and N the squae of the buoyancy fequency. Both citeia (1.1) and the geneal citeion (1.) have long been known to meteoologists (see e.g. Sawye 1947; van Mieghem 1951; Eliassen & Kleinschmidt 1957; Chaney 1973). The second condition in (1.) appeas in vaious disguises in the liteatue. The citeion fo symmetic stability of baotopic and baoclinic paallel shea flows in a otating system follow quickly by discading the cuvatue tem / in Φ and fo baoclinic flows in S as well.

3 iscous symmetic stability of cicula flows 173 The classical stability citeion (1.) fo baoclinic flows petains only to inviscid and adiabatic fluids and can be established in a vaiety of well-known ways. Fo instance, it can be established by including otation and statification in Rayleigh s fluid ing exchange agument fo cicula votices (Rayleigh 1916). The pessueless Lagangian displacement agument of Solbeg (1936) also leads to the stability condition. Fjøtoft (195) used an enegy method which, unlike in Rayleigh s o Solbeg s appoach, takes continuity, bounday conditions and pessue petubations into account. Using the fact that fo symmetic petubations the angula momentum o ciculation of individual fluid ings is conseved, Fjøtoft showed that the combination of total kinetic enegy associated with the azimuthal (swil) velocity and the potential enegy foms an effective potential enegy fo motions in the meidional z plane. This effective potential enegy depends only on the meidional displacement field. Fjøtoft futhe showed that the effective potential enegy fo a given basic flow is a minimum if the citeion fo stability (1.) is satisfied. If so, it follows that the incease in the total meidional kinetic enegy can be kept abitaily small if the initial velocity petubations and initial meidional displacements ae sufficiently small. Though a consideation of the time evolution of cetain volume integated quantities, Ooyama (1966) also found that in the lineaized dynamics the total meidional kinetic enegy and the meidional displacement field can be kept abitaily small if the classical condition fo stability (1.) is satisfied. If not, then initial petubations can be intoduced which lead to unbounded gowth of both integal quantities, i.e. thee will be instability. Fo the instability poof, Ooyama used a cleve constuction having noted that the assumption of the existence of nomalmodes solutions may be eoneous, i.e. geneally the lineaized dynamics allows fo a nomal-modes analysis only when the boundaies have a special shape, which diffes fo diffeent flows (see Høiland 196; Yanai & Tokiaka 1969). Thus, nomalmodes stability/instability o so-called exponential stability/instability can aely be expected to be established fo enclosed flows. If the question of the existence of nomal-modes solutions is disegaded, a nomal-modes analysis does howeve lead to the inviscid citeion fo stability. Fo baotopic flows in a homogeneous fluid this is shown in the Appendix, i.e. if (1.1) holds, thee will be nomal-modes stability. Fo baoclinic paallel shea flows in a otating system, Cho, Shephed & ladimiov (1993) as well as Mu, Shephed & Swanson (1996) showed that nonlinea stability in the sense of Lyapunov follows if (1.) is satisfied. They used the enegy-casimi methodology of Fjøtoft (195) and Anol d (1966). It equies the intoduction of a pseudo-enegy. The diffeence between the pseudo-enegy of the petubed flow and the basic unpetubed state is the distubance pseudo-enegy which is conseved in the fully nonlinea dynamics. If (1.) is satisfied, the distubance pseudo-enegy is a positive-definite functional of the finite-amplitude petubations which implies Lyapunov stability. In Kloosteziel & Canevale (7) it was shown how the pesumed symmety in the linea petubation poblem allows fo the constuction of an effective enegy E which also establishes stability in the sense of Lyapunov if the classical condition (1.) is satisfied. The distubance pseudo-enegy of Cho et al. (1993) educes in the smallamplitude limit to this effective enegy. The stability poofs of Fjøtoft (195), Ooyama (1966) and Cho et al. (1993) all depend on consevation of angula momentum o absolute velocity as well as density of individual fluid ings o ods. If viscosity and density diffusion ae included, these consevation laws no longe hold. But, the

4 174 R. C. Kloosteziel constuction of Kloosteziel & Canevale (7) allows fo the inclusion of viscosity and density diffusion (albeit just fo the lineaized dynamics). With thei appoach they ediscoveed McIntye s stability bounday fo double diffusive instability (McIntye 197). McIntye found this stability bounday with a nomal-modes analysis on an unbounded domain (appoximating a cicula flow by a ectilinea flow with constant vetical and hoizontal shea and embedded in a fluid with constant buoyancy fequency) in the limit of vanishing viscosity. This favouable compaison led us to expect that by using Kloosteziel and Canevale s appoach (Kloosteziel & Canevale 7) we might be able to extact useful infomation egading stability of moe ealistic viscous/diffusive flows. The fist step in this diection is taken in this pape by consideing abitay cicula flows in a homogeneous fluid. The plan of this pape is as follows. Section stats with the linea petubation equations. In.1 we constuct the effective enegy E which is conseved in the inviscid linea dynamics, i.e. de/dt =, wheet is time. The constuction is only valid fo flows fo which the (absolute) voticity Q is sign-definite, i.e. fo flows with Q eveywhee. The effective enegy is positive definite if the function / + f F() = d/d + / + f = / + f (1.3) Q is positive eveywhee and E then becomes a Lyapunov functional with which stability is investigated thoughout this pape. It follows that any flow that satisfies the inviscid classical citeion fo stability (Φ o F > eveywhee) is stable in the sense of Lyapunov in the inviscid poblem. In. we deive an uppe bound fo the gain in petubation enegy that may occu due to tansient non-modal gowth in the inviscid dynamics fo Lyapunov stable flows. In.3 we deive two conditions which guaantee that in the viscous dynamics de/dt at all times. The fist is that if thee is a constant α such that fo all, F() > and G(; α) d F d + α df d 1 (3 α)(1 + α)f, (1.4) then thee is Lyapunov stability. Stability is also guaanteed if eveywhee ( ) df/d F() > and 4 F. (1.5) Eithe (1.4) o (1.5) is sufficient: they need not both hold simultaneously. Also it is shown that if eithe of these conditions is met, the inviscid uppe bound on the gain deived in. emains valid fo any Reynolds numbe in the viscous dynamics. The citeia (1.4) and (1.5) ae the two notable esults in this pape. In the deivations of these esults we ignoe end-effects, i.e. we ignoe the fact that if thee is, fo example, a igid bottom and/o top, nea these boundaies the field should vanish if the no-slip condition applies. This inconvenience can be avoided (as is usually done) by imagining the fluid to be of infinite extent in the vetical o by assuming that such end-effects ae confined to a thin egion nea the boundaies. Also, at igid boundaies the velocity petubations ae equied to vanish, i.e. the no-flux and no-slip condition ae pescibed. In any case, the boundaies must also be ciculaly symmetic. Futhe we assume that the unpetubed flow can be consideed stationay.

5 iscous symmetic stability of cicula flows 175 In 3 we apply the theoy to Couette flow. We find with (1.4) that Couette flow is Lyapunov stable in the viscous dynamics if the classical inviscid citeion is satisfied. Synge (1938) poved with a nomal modes analysis that Φ() > is sufficient fo the viscous poblem (nomal-modes stability). Wood (1964) late gave a shot poof using an effective enegy integal like ous. Both poofs depended on that Φ 1 + constant/, so that when integating and diffeentiating simple powes of appeaed (see also Chandasekha 1961, who essentially epeats Synge s analysis). Hee we have howeve geneal conditions fo symmetic stability with which we can test the stability of any flow. In 4 we conside the Lamb Oseen votex. In 4.1 we find that in a nonotating envionment (f = ) we cannot pove stability with eithe (1.4) o (1.5). If placed in a otating envionment (f ) we discen between the cyclonic and anticyclonic case though the sign of the Rossby numbe Ro. Fo cyclones (Ro > ) we find in 4. Lyapunov stability fo a finite ange of positive Rossby numbes wheeas classical stability is guaanteed fo all Ro >. In 4.3 we show that fo anticyclones both (1.4) and (1.5) imply Lyapunov stability in the entie classically stable ange 1 <Ro<. Fo the anticyclone we find that fo any Reynolds numbe the petubation enegy can incease at most by a facto of. Fo the cyclone with lage Rossby numbe Ro = 6 the incease cannot exceed a facto of about 5. The maximum incease is smalle fo weake cyclones, i.e. fo smalle Rossby numbes. In 5 we conclude with a bief summay and discussion of the main esults and mention possible futue extensions of this wok. In the Appendix we show what the mathematical poblems ae that aise if a nomal-modes analysis is attempted with viscosity included. This could have been sufficient motivation fo the appoach we have taken in this pape.. Fomulation of the lineaized poblem Conside a steady cicula flow o votex v = () that is in cyclo-geostophic and hydostatic balance: + f = 1 P ρ, 1 P = g, (.1) ρ z whee p = P (, z) is the pessue, ρ is the constant density and g is the gavitational constant with gavity aligned with the axis of otation (along the z-axisassketched in figue 1) and f the Coiolis paamete epesenting the effect of otation in the dynamics. When the kinematic viscosity ν, stationay flows must satisfy 1 d d d d = (like Couette flow between two concentic cylindes). Nonetheless, when we pefom a stability analysis below in.3 with viscous effects included, we will teat the basic state as stationay. Whethe this leads to conclusions one can have confidence in, lagely depends on whethe the time scales of the evolution of the basic flow and those of the petubations ae well sepaated. One can also envision the possibility of intoducing an appopiate ciculaly symmetic extenal foce field acting in the azimuthal diection which endes the azimuthal v field stationay (see 5).

6 176 R. C. Kloosteziel Intoducing petubations u, v, w and p independent of the azimuthal angle θ and lineaizing about the balanced state (.1) we get ( ) ( ) t νδ 1 u + f v = 1 p ρ, (.) ( ) ( d t νδ 1 v + d + ) + f u =, (.3) ( ) t νδ w = 1 p ρ z, (.4) u = 1 (u) + w =, (.5) z whee Δ = 1 + and Δ z 1 = Δ 1. (.6) In what follows, it will also be useful to wite Δ 1 = 1 + z. (.7) Adding u (.) + w (.4) we get 1 ( u + w ) ( ) + f uv = u p/ρ + ν(uδ 1 u + wδw), (.8) t whee = e + e z z, u = e u + e θ v + e z w. With (.3) it follows that ( d u = ( v/ t νδ 1 v)/ d + ) + f = ( v/ t νδ 1 v)/q (.9) povided that the (absolute) voticity Q defined in (1.1) vanishes nowhee. Thee will be difficulties, fo example, if f =and is potential flow (i.e. 1/), as found outside a spinning cylinde which is placed in a lage quiescent basin. In that case ou analysis fails because then Q = so that (.9) is meaningless. Assuming that Q eveywhee, we can substitute (.9) in (.8) and get 1 [ u + w + Fv ] = u p/ρ + ν(uδ 1 u + wδw + FvΔ 1 v) (.1) t with F() defined in (1.3)..1. Inviscid dynamics If we set ν = and integate (.1), we find that de = with E = 1 [ u + w + Fv ] d (.11) dt povided that div(up/ρ)d =.d =π d dz stands fo the volume integal ove the domain. This will hold if the flow is eithe enclosed by igid boundaies whee u n = with n the unit vecto nomal to such boundaies (no flux condition), o can be in a domain unbounded in one o moe diection, in which case we equie

7 iscous symmetic stability of cicula flows 177 that u vanishes fast enough fo if unbounded in the adial diection, o fo z ± if unbounded in the vetical. If thee is no inteio bounday, things must of couse be well behaved at the oigin =. Fom hee on we shall call E the effective enegy. If / + f eveywhee the effective enegy can also be witten as E = 1 ] [u + w (/+ f ) + v d. (.1) Φ We mentioned this esult (without a deivation) aleady in Kloosteziel et al. (7). Clealy E is a positive-definite functional if the inviscid classical condition Φ> is satisfied eveywhee in the domain because then also F > eveywhee. This establishes Lyapunov stability in the inviscid dynamics, i.e. the petubation enegy E, u [ u d] 1/, w and v can be kept abitaily small at all times by choosing the initial petubations small enough. To see this, note that if F > eveywhee, it follows that if at some initial time (say t = ) small petubations ae intoduced, then at all times (subscipts indicate t =) E(t) =E = 1 [ ] u + w + Fv d > and (.13) 1 (u +w )d E, 1 min {F} v d 1 Fv d E. (.14) The last inequality implies 1 v E d min {F}. (.15) Fom hee on min{ } and max{ } stand fo the minimum and maximum of the function in the domain. The total petubation enegy E is E = 1 [ u + w + v ] d. (.16) Fo convenience we have dopped the constant density ρ hee. It easily veified that E E if min {F} 1 and min {F} (.17) E E if min {F} 1. Hence E can be kept abitaily small if F > eveywhee povided that E <. Finite E is guaanteed if F < eveywhee which means that Q = cannot be allowed. The petubation enegy evolves accoding to de dt = [( d d ) uv ] d + ν (uδ 1 u + vδ 1 v + wδw)d. (.18) If the classical citeion fo stability (Φ >of > eveywhee) is satisfied and ν =, clealy one would be had pessed to establish stability with (.18). But, (.17) shows that thee will then always be Lyapunov stability... Tansient gowth Although the petubation enegy can be kept abitaily small by taking the initial petubations u,v and w small enough, thee can be tansient gowth de/dt >

8 178 R. C. Kloosteziel fo some peiod of time. Much ecent eseach has focused on finding optimal petubations fo a vaiety of flows that lead to the geatest possible tansient amplification of initial petubations. This tansient gowth phenomenon can occu in any system whee the opeatos that descibe the lineaized dynamics ae nonnomal, i.e. not self-adjoint so that eigenmodes (as in nomal-modes analysis) ae not mutually othogonal. Even in systems that ae exponentially stable (nomal-modes stable), lage gowth fo some finite time-inteval is sometimes possible fo lage Reynolds numbes (see e.g. Faell 1988, Butle & Faell 199, Tefethen et al. 1993, Padeep & Hussain 6, Schmidt 7). An uppe bound fo the so-called gain G(t) =E(t)/E can be detemined as follows: Fist note that E 1 [ u + w +max{f} v 1 ( ] d = u + w +max{f} v ). (.19) Next note that E u + w +max{f} v max {F} when max {F} 1, E u + w + v (.) 1 when max{f} 1. Dividing both sides of the inequalities in (.17) by E yields with (.) E(t)/E = G(t) G max =1/min {F} when max {F} 1, (.1a) =max{f} when 1 min {F}, (.1b) =max{f} /min {F} when min {F} 1 max {F}. (.1c) This uppe bound G max fo the gain is valid fo any classically stable and theefoe Lyapunov stable flow in the inviscid dynamics. If F 1 eveywhee, then G max slightly exceeds unity. If one imagines, fo example, that F = 1 (take f =and solid-body otation () =Ω)thenE = E and G max = 1 accoding to (.1c). Thee is neithe tansient gowth no decay: both de/dt =andde/dt =. One may wonde whethe these uppe bounds ae shap o not. Conside fist the case < F 1 so that accoding to (.1a) the uppe bound is G max = 1/min {F}. Imagine that initially thee ae meidional velocity petubations u,w while v =.ThenE =(1/)( u + w )=E. Assume that late the meidional velocity petubations vanish and instead thee is a highly concentated azimuthal petubation velocity field v cented about the position = min whee F( min )= min {F} 1. Then E(t) =(1/) v(t) and by consevation of E we have E(t) (1/)min {F} v(t) = E = E o E(t) E /min {F}. ThenE(t)/E 1/min {F} which is (.1a). Physically this scenaio is plausible if the initial u, w fields ae concentated about min. In the inviscid dynamics we expect that the uppe bound (.1a), although unattainable, is quite shap and that maximum amplification is found fo initial meidional velocity petubations located in a naow egion whee F() attains it minimum. The uppe bound (.1b) fo cases with F 1 follows likewise by imagining that initially thee is an azimuthal petubation field v highly concentated about = max whee F( max )=max{f} 1. If u = w = we then have E =(1/) v and E (1/)max {F} v. If at a late time v = and thee ae non-zeo meidional velocity petubations then E(t) =(1/)( u(t) + w(t) )=E(t). But consevation of E implies E(t) =E(t) max {F} E so that E(t)/E max {F}. This suggests

9 iscous symmetic stability of cicula flows 179 that the uppe bound in (.1b) could be appoached by choosing an initial v field that is confined to a naow egion about = max. If max {F} > 1andmin{F} < 1, the uppe bound G max =max{f} /min {F} in (.1c) is found by imagining that initially thee is a v field concentated in the egion about max while u = w = and that at a late time again u = w = while the v field is then concentated aound = min.thisisnotaplausiblescenaio: it is unlikely that an initial v field concentated in one egion evolves accoding to (.) (.4) towads a v field that is concentated in a diffeent egion. Hence fo flows with min {F} 1 max {F}, G max in (.1c) is pobably athe consevative..3. iscous dynamics We will now deive citeia fo viscous flows that guaantee that E is positive definite while at all times de/dt. With the assumption that we can teat as stationay, we have de dt = ν (uδ 1 u + wδw)d + ν FvΔ 1 v d. (.) div(up/ρ)d =. At boundaies we equie the Again, this is only valid if petubations u, v, w to vanish and in any infinite diection they ae again equied to vanish apidly enough. Then (.) is valid and though patial integation it follows that the fist tem on the ight-hand side in (.) becomes ν (uδ 1 u + wδw)d = ν ( 1 u + w ) d, (.3) whee a b = a ( b) + a ( z b). The tem (.3) would also aise as a pat of the usual viscous dissipation of the petubation enegy E. Let us evaluate the second tem on the ight-hand side of (.). The pat involving the z opeato in Δ 1 simply gives the tem ν Fv z v d = ν F( zv) d. The pat involving the deivatives can be evaluated in two ways, (A) uses the fom Δ 1/ with Δ as in (.6) while (B) uses the opeato as in Δ 1 in (.7). Assuming F to be continuous and twice diffeentiable, we find though patial integation: (A) Fv ( 1 (B) Fv Thus we get (A) (B) de dt de dt v v ) d = 1 v d = F = ν + ν = ν + ν + 1 ( 1 v F 3 [ ( v ) ] ( v ) + d 1 d d df d (v) d, ) d d d 1 df d (v) d. ( 1 u ) + w d ν F v d ( d F d + 1 df d F ) v d, (.4) ( 1 u ) + w d ν F 1 v d ( d F d 1 ) df v d. (.5) d

10 18 R. C. Kloosteziel Fom this it follows that a sufficient condition fo stability is that F() > and (A) d F d + 1 df d F o (B) d F d 1 df, (.6) d because then E =(1/) (u + w + Fv d) is positive definite and at all times the ight-hand side in (.4) o (.5) is negative o zeo and thus de/dt. But, this can be impoved as follows: Conside (.4) c (.5), with c 1a constant: (1 c) de { = (1 c)ν 1 u ( ) } v + w + F d dt z + ν ν { (1 c) d F d { F +(1+c)1 df d [ ( v ) ] ( v ) (1 c) + } v d c v } v d. (.7) The tem { } in the last integal on the ight-hand side in (.7) equals { ( v ( c ) ) ) } v (v ) (1 c) + (1 c. 1 c (1 c) We divide both sides of (.7) by (1 c) andset α = 1+c ) so that (1 c = 1 1 c (1 c) (3 α)(1 + α) and c c 1 = 1 (α 1). The effective enegy equation then becomes { de = ν u ( ) } ( v v + w + F d ν F dt z 1 ) (α 1)v d + ν { d F d + α df d 1 (3 α)(1 + α)f } v d. (.8) Theefoe a sufficient condition fo stability of a viscous flow with espect to abitay symmetic petubations is that, fo some constant α, (1.4) is satisfied eveywhee. In (1.4) only the ange 1 α 3 is useful since only in that ange (3 α)(1 + α). Fo α = 1 this is condition (A) in (.6), while (B) is found fo α = 1. Stability also follows if F > andif 1 { d F d + α df d 1 (3 α)(1 + α)f } ( v v d F 1 ) (α 1)v d. (.9) If F>wecan substitute v = ṽ/ F in (.4), (.5) o (.8). Then v = 1 ṽ F 1/ 1 ( ) df/d ṽ, F 3/ and if this is substituted in the integal F v d that appeas in (.4) one gets ( ) { ( ṽ ) ṽ F v d= d+ ṽ df ṽ z F d + 1 ( ) } df/d ṽ d. 4 F

11 iscous symmetic stability of cicula flows 181 Next we use that ( ) ṽ df ṽ F d d = 1 ṽ df ddz + 1 ( ) ṽ d df d. F d d F d Assuming v and theefoe ṽ to vanish at boundaies o apidly enough fo lage o at =, the integal / ( ) d dz = and we find that (.4) becomes { de = ν 1 u ( ) } ṽ + w + d dt z ν { ( ṽ ) 1 4 [ (df/d F ) 4 ] } ṽ d. (.3) Equations (.5) and (.8) also take this fom afte the substitution v = ṽ/ F. Hence stability is also guaanteed if (1.5) holds eveywhee. One might expect that (1.5) is a less consevative condition than (1.4) since (1.5) follows fom combining the second negative-definite integal on the ight-hand side of (.8) with the last integal containing G(; α) which we defined in (1.4). But, as we will see in the next section, this is not necessaily tue. In any case, if eithe (1.4) o (1.5) is satisfied, de/dt =onlywhen u, w, v and v = eveywhee in the domain. If the domain is eithe confined in the vetical o in the hoizontal by igid boundaies whee the petubations vanish, this implies that u = v = w = so that lim t E = and lim t E =. Hence the flow would then be foced back towads the basic flow () and the flow is asymptotically stable. If eithe (1.4) o (1.5), o both, ae satisfied eveywhee, the uppe bounds G max on the gain due to possible tansient gowth emain (.1a) (.1c) because then de/dt at all times so that instead of (.17) we have E(t) E(t) min {F} E if min {F} 1, min {F} (.31) E(t) E(t) E if min {F} 1. Division by E and using (.) then again yields (.1a) (.1c). Fo small Reynolds numbes Re, these uppe bounds ae expected to be consevative because the evolution equations (.8) and (.3) indicate that the effective enegy E will diminish apidly fo small Re if eithe (1.4) and/o (1.5) is satisfied eveywhee. 3. Couette flow The well-known Couette flow of a viscous fluid between otating coaxial cylindes has the velocity distibution μ η () =A + B/ with A = Ω 1 1 η, B = Ω 1R 1 μ 1 (3.1) 1 η and the paametes μ = Ω /Ω 1 and η = R 1 /R. R 1 and R ae the adius of the inne and the oute cylinde which otate with angula velocity Ω 1 and Ω, espectively (see Dazin & Reid 1981). The Rayleigh disciminant Φ() (with f = ) is positive thoughout the domain R 1 R if

12 18 R. C. Kloosteziel μ>η. This means that fo classical (inviscid) stability the cylindes must otate in the same diection and (R )R > (R 1 )R 1. (3.) We have F() =1+ B ( ) 1 μ A =1+ 1 (3.3) μ η (/R 1 ) and F() > foall [R 1,R ] if the classical citeion (μ >η ) is satisfied (note that Φ() =4A F() fo Couette flow). Like evey flow that satisfies the classical condition, in the inviscid dynamics Couette flow is Lyapunov stable. The second condition in (1.4) is afte substitution of (3.3) G(; α) =(6 α) B 1 A 1 ( 1 (3 α)(1 + α) 4 + B ) 1. (3.4) A 4 This will be satisfied fo α =3,i.e.foα = 3 the last tem in (.8) vanishes identically fo all >. Hence Couette flow is also Lyapunov stable in the viscous dynamics if the inviscid classical citeion is satisfied. Fo Couette flow (1.5) is too consevative: we find that the second condition in (1.5) is only satisfied when μ< η. This implies that the classical condition plus the second condition in (1.5) equie that the inne and oute cylinde otate in the same diection and that (R )R > (R 1 )R 1 but (R )R < (R 1 )R 1 ( (R /R 1 ) 1 ). (3.5) Thus (1.5) only poves Lyapunov stability of a subset of the classically stable Couette flows wheeas (1.4) poves stability fo all classically stable flows. Fo diffeent flows the convese may be tue, i.e. (1.5) may sometimes be less estictive than (1.4). This is shown in the next section with an example. Thee ae two special cases wothwhile mentioning. The fist is that of whee the inne cylinde is taken out. Then = Ω, which is simply solid-body otation as found inside a otating cylinde afte a sufficiently long spin-up time. Then Φ =4Ω and F = 1. This is theefoe inviscidly stable to symmetic petubations because the effective enegy is positive-definite, but also Lyapunov stable in the viscous dynamics since eithe (.4) o (.5) show that de/dt andalsode/dt. The second case is the limit R and Ω =(μ =andη = ). The esult would be pue potential flow = Ω 1 R1 / outside a spinning cylinde. This has Φ = eveywhee because the voticity Q is zeo and F is undefined. As mentioned in, ou appoach cannot be used in this case. 4. Lamb Oseen votex The so-called Lamb Oseen votex o Gaussian votex has a velocity distibution () and coesponding voticity ω given by () = ω L [ ( 1 exp /L )], ω() = d (/L) d + = ω exp( /L ). (4.1) The adius has been non-dimensionalized with an abitay length scale L and ω is the peak voticity found at =. and ω ae shown in figue (a). Unlike Couette flow, in a feely evolving viscous fluid this is not a steady state solution of the Navie Stokes equations and and ω evolve accoding to t = νδ 1 and

13 iscous symmetic stability of cicula flows 183 (a) 1. (b) 3 ω/ω L d /d.5 /ω L 1 4/(/L) /L c /L /L Figue. (a) The voticity ω (stippled line) and velocity (solid line) of the Lamb Oseen votex given by (4.1) non-dimensionalized with ω and ω L, espectively, as a function of /L. Peak voticity is ω at =andl is abitay. The non-dimensional peak velocity is max /(ω L).45 at max /L (b) Gaph showing F given by (4.) (thin line), (df/d) /F non-dimensionalized with L (thick line) and the cuve 4/(/L) (stippled line) as a function of /L. Fo> c /L 1.79 (indicated by ) the citeion (1.5) is not satisfied. In (a) the symbol is also shown at the position c /L. t ω = νδω. If (4.1) defined and ω at some time t =, the time evolution is (, t) = ω (t)l(t) [ ( 1 exp /L(t) )], ω(, t) =ω (t) exp ( /L(t) ) (/L(t)) with ω ω (t) = 1+(νt/L ()), L(t) = L () + νt. But, we will teat the flow as stationay and only conside and ω defined in (4.1) Non-otating system In a non-otating system (f = ) the Rayleigh disciminant Φ =(/) ω>foall and F() = / /L ) 1 ω =exp( 1 (4.) (/L) fo all, as shown in figue 3(b). The smallest value F =1isfoundinthe limit. Thus the effective enegy is positive definite. But Lyapunov stability with espect to symmetic petubations in the inviscid dynamics cannot be established. The eason is that as we have F.Thisisduetothefactthatfo lage the voticity Q = ω vanishes exponentially fast (the votex gets eve close to potential flow 1/ so that Q ). An uppe bound G max fo the possible gain due to tansient gowth in the inviscid dynamics cannot be detemined with (.1b) unless the flow is teminated flow at some finite = c. A finite G max = F( c ) can be calculated but it can be made abitaily lage by inceasing c. The fact that G max is not finite on the infinite domain is pehaps not supising since it is known that potential flow can suppot unbounded algebaic gowth (see 5 fo a bief discussion). Fo the viscous dynamics we need to detemine whethe (1.4) o (1.5) can hold eveywhee. In figue 3(b) it is seen that fo /L > c /L 1.79 the citeion (1.5) does not hold, i.e. the second condition in (1.5) is violated. Between =and = c (1.5)

14 184 R. C. Kloosteziel (a) 4 (b) 1..8 L α = 1 α = c /L.4 α = /L α Figue 3. (a) The function G(; α) given in (1.4) and non-dimensionalized with L,fothe Lamb Oseen votex and α = 1, andα =.Foα = 1 andα = 3 (not shown) G fo all. Foα = we have G > foall/l > c /L (b) Gaph showing the citical adius c /L as a function of α [ 1, 3]. The geatest value is found fo α. with c /L Thee ae no α values fo which G foall so (1.4) is neve satisfied on the unbounded domain. The symbol is at the position /L = c /L 1.79 fom figue 3. is satisfied. If the flow was teminated at = c, the flow would be stable but on an unbounded domain Lyapunov stability cannot be established with (1.5). In figue 3(a) weshowg(; α), defined in (1.4), fo a few values of α. In each case thee is a finite value = c with G > when> c. In figue 3(b) weshowthe citical value c /L as a function of α [ 1, 3]. Thee ae no α values fo which (1.4) is satisfied eveywhee. Futhe, the egion whee (1.4) can be satisfied is confined to a fa smalle ange of values than (1.5): the widest egion fo which Lyapunov stability can be established accoding to (1.4) is /L [, 1.19] wheeas (1.5) poves stability fo the ange /L [, 1.79]. On a adially unbounded domain we theefoe cannot pove Lyapunov stability. 4.. Rotating system: cyclones We now add the Coiolis foce to the dynamics (f ) and define the Rossby numbe fo the Lamb Oseen votex as Ro = ω /f.ifro > this is a cyclonic votex, if Ro < it is an anticyclonic votex. The Rayleigh disciminant non-dimensionalized with f is Φ()/f = ( Ro/(/L) [ 1 exp ( /L )] +1 )( Ro exp( /L )+1 ), (4.3) while [ F = Ro/(/L) 1 exp ( /L )] +1. (4.4) Ro exp( /L )+1 Fo all Ro ( 1, ) itcanbeshownthatφ() > foall (see Canevale et al. 1997). Fo the cyclone (Ro > ) this is easily seen with (4.3). Futhe, fo all Ro > 1 we find that F > 1/ foall. Thus, when Ro > 1 the effective enegy is positive definite and the votex is theefoe Lyapunov stable in the inviscid dynamics fo all Rossby numbes Ro > 1. The smallest value F =1/ is found in the limit Ro 1and. The limiting case Ro 1 is singula in that the absolute voticity Q = ω + f tends to zeo in the limit sothatf becomes undefined thee (emembe F =(/ + f )/Q). Fo the viscous dynamics, we fist conside the cyclone case Ro >. In figue 4(a) we show F fo a few positive Rossby numbes. Geneally fo all Ro > we have

15 iscous symmetic stability of cicula flows 185 (a) Ro = 5 Ro = 1 Ro c = (b) 1..5 Ro c = 6 L Ro = 5 d /d Ro = 1 4/(/L) /L /L Figue 4. (a) F as a function of /L fo the cyclonic Lamb Oseen votex with Ro = ω /f =1, 6 (thick line) and Ro = 5. Fo any Ro > we have F 1foall. (b) (df/d) /F fo Ro =1, 6 and 5 and non-dimensionalized with L (solid lines) and the cuve 4/(/L) (stippled line) as a function of /L. Fo<Ro Ro c 6 the citeion (1.5) is satisfied fo all (numbes ae appoximate: 6 <Ro c < 6.1). Fo Ro = Ro c the cuve L (df/d) /F touches the cuve 4/(/L) at /L.5 (thick line). Fo Ro > 6 thee is a ange of values in which the second condition in (1.5) does not hold. Fo all Ro < 6 the cyclone is Lyapunov stable accoding to (1.5). (a) 3 α = 1 α = 3 (b) 3 max L Ro = 6 L 3 α =.6 Ro c = /L 1 Ro c = α Figue 5. (a) The function G(; α) given in (1.4) and non-dimensionalized with L fo the cyclonic Lamb Oseen votex with Ro = Ro c =1andα = 1,.6 (thick line) and α =3.(b) Numeically detemined non-dimensional maximum of G fo Ro = Ro c =1andRo =6as a function of α [ 1, 3]. Fo Ro > Ro c = 1 the maximum is positive fo all α which means that thee is always a ange of values whee the second condition in (1.4) is violated. Fo α.6 the maximum is zeo, as shown in (a). Fo all Ro < Ro c thee is always an α fo which G foall. Accoding to (1.4) this guaantees Lyapunov stability. F 1foall. Numeically we find that fo <Ro Ro c 6 the citeion (1.5) is satisfied. This is shown in figue 4(b). Fo Rossby numbes geate that Ro c 6 the second condition in (1.5) is violated fo some ange of values (the pecise value lies between 6 and 6.1). Thus, (1.5) establishes Lyapunov stability of the cyclonic Lamb Oseen votex only in the ange Ro (, 6) in the viscous dynamics wheeas stability is guaanteed fo all Ro > in the inviscid dynamics. The stability citeion (1.4) is satisfied fo a smalle ange of positive Rossby numbes, i.e. only fo <Ro 1. In figue 5(a) weshowg(; α) foro = Ro c 1 and a few values of α as indicated. Fo α.6 wefindthatg foall. This

16 186 R. C. Kloosteziel (a) 1. Ro =.1 (b) 1 4/(/L) Ro =.5.75 Ro =.99 1 Ro =.99 L d /d 1 Ro = /L Ro = /L Figue 6. (a) F as a function of /L fo the anticyclonic Lamb Oseen votex with Ro = ω /f =.9999,.9, 1/ and Ro = 1/1. Fo all Ro > 1 we find that F 1/ foall. (b) (df/d) /F non-dimensionalized with L (solid lines) and the cuve 4/(/L) (stippled line) as a function of /L. Fo 1 <Ro< the citeion (1.5) is satisfied fo all. This is illustated hee with the examples Ro =.9 (thick line) and Ro = 1/ (thin line). Also fo Ro =.9999 is L (df/d) /F < 4/(/L) fo all (not shown). is seen in figue 5(a) whee fo α =.6 G touches the zeo level at some between =1and =, becomes negative again and then asymptotically G fo. In this case with Ro 1, fo all α.6 we find that G > fo some ange of values. This is illustated in figue 5(a) with two examples. In figue 5(b) we show max {G} as a function of α. FoRo = Ro c 1 we see that max {G} = just fo α.6, while fo all othe α the maximum is positive. Fo Ro 1 we find max {G} > foallα and the second condition in (1.4) can theefoe not be satisfied fo any α when Ro 1. This is illustated in figue 5(b) with Ro = 6. Fo all Rossby numbes <Ro<Ro c 1, thee is always an α [ 1, 3] fo which G fo all (in each case G as ). Thus, with (1.5) we find the widest ange of positive Rossby numbes ( <Ro 6) fo which we can pove Lyapunov stability in the viscous dynamics. Contay to the case of the votex in a non-otating envionment, we can detemine the uppe bound on the gain G max. Since fo any finite positive Rossby numbe 1 F <, accoding to (.1b) we have G max =max{f}. Figue 8(a) showsg max as a function of the Rossby numbe in the (viscously) stable ange <Ro 6. Fo the lagest Rossby numbe Ro = 6 we find that G max 5.5. Since this uppe bound is valid fo any Reynolds numbe, including the limit Re,thisisinmaked contast with the esult of Padeep & Hussain (6) who studied the Lamb Oseen votex in a non-otating system. They showed that thee can be significant tansient gowth of axisymmetic petubations well outside the coe of the votex. A 1-fold incease in the total petubation enegy E (a gain G =1 ) was found even fo a modest Reynolds numbe Re = 5, although the flow is nomal-modes stable. The diffeence between the otating and the non-otating case is futhe discussed in Rotating system: anticyclones In the inviscid dynamics the anticyclonic Lamb Oseen votex is classically and Lyapunov stable fo all Rossby numbes in the ange 1 <Ro<. In figue 6(a) we show F fo a few negative Rossby numbes. Fo Ro 1 (illustated in figue 6a with Ro =.9999) the smallest possible value F =1/ is appoached nea =.The

17 iscous symmetic stability of cicula flows 187 (a) α =.57 α =.34 (b) α =.74 α =.66 L.5.5 α = 1 Ro = /L 1. α = 1 Ro = /L Figue 7. (a) The function G fo the anticyclonic Lamb Oseen votex with Ro =.99 and α values as indicated. Fo all α [.57,.34] we find that G foall. Fothetwo bounding values of this ange G is shown as a thick line. (b) Sameas(a) but fo Ro = 1/ and α values as indicated. Fo all α [.74,.66] we have G foall. In both cases the anticyclonic Lamb Oseen votex is Lyapunov stable accoding to (1.4). Geneally fo any Rossby numbe 1 <Ro<theeisaangeofα values within the ange [ 1, 3] that yield G foall while G fo. The naowest ange is found in the limit Ro 1. Fo Ro =.9999 it is to thee significant digits the same as shown in (a), i.e. fo.57 α.34. (a) 5 (b). 4 3 G max 1.5 G max Ro Ro Figue 8. Uppe bound G max fo the gain G(t) =E(t)/E() fo (a) the cyclonic Lamb Oseen votex fo the Lyapunov stable ange of Rossby numbes <Ro 6 with G max =max{f} and (b) fo the anticyclonic Lamb Oseen votex in the Lyapunov and classically stable ange 1 <Ro< with G max =1/min {F}. FoRo = 6 we have G max 5.5 in (a) while fo Ro 1in(b) G max. The bounds ae valid fo any Reynolds numbe. second condition in (1.5) is satisfied fo all Rossby numbes in the classically stable ange 1 <Ro<. This is illustated in figue 6(b) with the examples Ro =.9 and Ro = 1/. A logaithmic scale is used along the vetical axis because of the lage diffeences between the peak values of (df/d) /(F) as the Rossby numbe vaies. So, contay to the cyclonic case, (1.5) poves Lyapunov stability of the anticyclonic Lamb Oseen votex in the viscous dynamics fo all Rossby numbes in the classically stable ange. We find that (1.4) also poves stability fo the entie classically stable ange: fo all Rossby numbes in this ange ( 1 <Ro<) thee is always an α [ 1, 3] fo which G foall. Figue 7 shows two examples: in figue 7(a) the Rossby numbe is Ro =.99, in figue 7(b) Ro = 1/. In both cases thee is a negative and positive

18 188 R. C. Kloosteziel α (indicated by the thick lines) fo which G = at some finite. Foallα in between these two extemes, G < fo all finite and only asymptotically G as. Fo Ro =.99 Lyapunov stability follows fo.57 α.34 (see figue 7a), fo Ro = 1/ stability fo.74 α.66 (see figue 7b). The smallest ange of α values with which Lyapunov stability follows is found in the limit Ro 1. The bound G max on the gain is shown in figue 8(b). Since < F 1 in this case, accoding to (.1a) G max =1/min {F}. Intheentieange 1 <Ro<, G max neve exceeds a value of. Hence just as in the cyclonic case, no significant tansient gowth can be expected fo the anticyclonic Lamb Oseen votex with Rossby numbes in the stable ange, no matte how lage the Reynolds numbe is. 5. Summay and discussion In this pape we have fist shown that if the inviscid classical citeion fo symmetic stability is satisfied (F() > oφ() > eveywhee), the effective enegy E defined in (.11) o (.1) is a Lyapunov functional. It establishes Lyapunov stability in the inviscid dynamics fo abitay cicula flows with espect to ciculaly symmetic petubations. In the inviscid dynamics de/dt = and we showed that G max in (.1a) (.1c) povides an uppe bound fo the amplification of petubation enegy which may occu due to tansient non-modal gowth in the inviscid dynamics. The development is somewhat simple but othewise analogous to the case of paallel shea flows in statified fluids as discussed in Kloosteziel & Canevale (7). Next we have deived two novel citeia fo Lyapunov stability of viscous cicula flows with espect to symmetic petubations, i.e. (1.4) and (1.5). In both citeia we fist find the equiement fo classical inviscid stability, i.e. Φ>oF > which guaantees that E is positive definite. The additional conditions guaantee that in the viscous dynamics de/dt at all times. This implies that the uppe bound G max fo the gain in the inviscid dynamics emains valid fo the viscous dynamics. If the second conditions in (1.4) and (1.5) ae not satisfied in some ovelapping egions, thee is the possibility that the effective enegy gows fo some peiod of time, i.e. E(t) > E and then (.1a) (.1c) maynotbetue. The theoy has been applied to a few examples. Fo othe types of votices o confined cicula flows than discussed in this pape, one must apply both (1.4) and (1.5) to see if the flow is povable stable. If a flow is chaacteized by a vaiable paamete like the Rossby numbe in otating systems o some paamete which detemines the velocity distibution and is povable stable with both (1.4) and (1.5), the esults must be compaed to see which of the two citeia poves stability fo the boadest ange of the paamete(s). It is impossible to pedict apioi. Fo Couette flow we established in 3 with (1.4) Lyapunov stability fo the entie classically stable ange by setting α = 3. A nomal-modes analysis by Synge (1938) and an enegy method by Wood (1964) had aleady established this, but both studies wee specifically aimed at Couette flow. In this study it quickly followed with the geneal condition (1.4) which can be applied to any flow fo which Q eveywhee. In 4.1 we found it impossible to pove Lyapunov stability of the Lamb Oseen votex in a non-otating system (f = ), both in the inviscid and the viscous dynamics. Fo the inviscid case this is not supising because fo lage the Lamb Oseen votex appoaches potential flow, i.e. accoding to (4.1) fo lage we have appoximately () ω L / so that Q. As noted by Miyazaki & Hunt (), potential flow suppots unbounded algebaic gowth. This is easily seen by setting ν =andf = in (.3) so that v/dt =when 1/. Hence fo potential flow v(, z, t) =v (, z)

19 iscous symmetic stability of cicula flows 189 emains unchanged. This can dive gowth of the azimuthal voticity ω θ in the lineaized dynamics accoding to ω θ t = t [ u z w ] = v z (5.1) povided that v / z does not vanish eveywhee. Theefoe the petubation kinetic enegy can gow without bounds if v / z. Padeep & Hussain (6) also noted that in the inviscid case the Lamb Oseen votex can expeience amplification of petubation enegy that becomes unbounded as the optimal petubations ae initiated at eve inceasing distances fom the votex axis. Ou thought expeiment in., which showed how G max =max{f} in (.1b) might be appoached, agees with thei obsevation that fo initial petubations concentated at eve lage, an eve inceasing gain is expected. In the viscous case, Padeep & Hussain (6) calculated a maximal gain of about G max =1 fo Re = 5 which will continue to incease indefinitely with inceasing Reynolds numbe. With ou appoach we could not establish a finite uppe bound fo the gain because fo lage both (1.4) and (1.5) ae not satisfied so that (.1a) (.1c) may be false. In 4. (1.5) poved stability of the cyclonic Lamb Oseen votex in the viscous dynamics fo a finite ange of positive Rossby numbes <Ro 6 wheeas thee is classical stability fo all Ro >. With (1.4) stability followed fo a smalle ange of Rossby numbes. Fo <Ro 6 we found 1 <G max 5.5. Ou esults imply that even in the case of athe weak otation (lage positive Ro but Ro < 6 o small otation numbe 1/Ro) a numeical seach fo significant gain (seveal odes of magnitude) in petubation enegy would be futile no matte how lage the Reynolds numbe is. Fo Ro > 6 we do not expect a sudden tansition to fa lage tansient gowth than fo Ro 6 because thee is no destabilizing mechanism pesent like the doublediffusive mechanism fo baoclinic flows mentioned in the intoduction. Since eve inceasing Ro can be intepeted as eve weake otation, the computations by Padeep & Hussain (6) fo the non-otating case suggest that as Ro tends to infinity, fo finite Reynolds numbes the maximum gain gadually appoaches some finite value G max (Re) andg max as Re. Fo the anticyclonic Lamb Oseen votex we found in 4.3 that eithe condition (1.4) o (1.5) established stability in the viscous dynamics fo the entie classically stable ange 1 <Ro< with 1 <G max <. We must note that a study of the evolution of an integal quantity like the effective enegy E cannot pedict what kind of petubations, chaacteized by spatial stuctue and elative amplitudes of the thee petubation velocity components, lead to maximal gain in the petubation enegy E. This can only be detemined with a numeical seach pocedue as descibed by Padeep & Hussain (6). Fo inviscid flows likely candidates ae petubations concentated about the maximum o minimum of F but fo viscous flows futhe analysis is equied. The modest uppe bounds fo the possible gain in the case of the Lamb Oseen votex in a otating envionment can be undestood with simple model which captues the inviscid behaviou. The model equations ae u t = ( ) + f v, v t ( d = d + ) + f u = Qu, (5.)

20 19 R. C. Kloosteziel which follow fom (.) and (.3) by ignoing pessue petubations. If we solve (5.) fo some position (, z), the geneal solution is ( ) ( ) ( ) u(t) =u cos Φt + v F sin Φt, v(t) =v cos Φt u sin ( Φt ). F (5.3) If = and f one has F = 1 and Φ = f and (5.3) descibes simple inetial oscillations. The effective enegy fo this model is conseved, i.e. E(t) = (1/) ( u (t)+fv (t) ) =(1/) ( u + ) Fv = E but the kinetic enegy is not unless F = 1. If one chooses a location whee F > 1 and takes u =,thene =(1/)v but at time t =/( Φπ) we have E(t) =(1/)Fv and the gain is theefoe F > 1. If F < 1 a gain of 1/F > 1 is found by taking v =. This is essentially a mathematical fomulation of ou thought expeiment at the end of. which made it plausible that the uppe bounds (.1a) and (.1b) ae faily shap in the inviscid dynamics. A Taylo-seies expansion yields ( ) u(t) =u + v + f t 1 u Φt +..., v(t) =v u Qt 1 v Φt +... (5.4) The highe ode tems (popotional to t 3, t 4, etc.) vanish in the limit Q in which case also Φ. The algebaic gowth fo potential flow 1/ when f =, which means Q =, is ecoveed povided that v.whenf,theu and v field ae coupled though the Coiolis foce and geneally this leads to oscillatoy behaviou. Fo lage in the Lamb Oseen votex Q while F 1 and lage amplification is possible but the gowth is vey slow because Φ. Fo the Lamb Oseen votex in a otating envionment we have Φ>foallRo > 1. Fo the cyclone Φ f wheeas fo the anticyclone Φ f (1 Ro ). In the context of the simple model this means that fo the cyclone the time scale T max fo the tansient amplification is faste than 1/f,i.e.T max < 1/f and fo the anticyclone T max < 1/(f (1 Ro )). A model with Rayleigh damping added to (5.) has been discussed by Padeep & Hussain (6, equation (4.)). This is a easonable model fo the viscous dynamics. It is clea that with damping the maximum gain will be smalle and occu at an ealie time than in the undamped dynamics. Fo ou simple model this means that if damping is added, the time scale T max emains smalle than 1/f fo the cyclonic Lamb Oseen votex andsmallethan1/(f (1 Ro )) fo the anticyclone. As is quite common in viscous stability studies, we assumed that the basic flow () is stationay in the viscous dynamics. This is tue fo Couette flow, solid body otation and potential flow but not, fo example, fo the Lamb Oseen votex. This was not mentioned by Padeep & Hussain (6) and vaious othes peceding them with studies of viscous tansient gowth. If we imagine the pesence of a ciculaly symmetic foce field acting in the azimuthal diection, i.e. a foce F θ () = ρνδ 1 () then any flow would be stationay. Fo such a foced-dissipative system, ou esults emain valid. But fo feely evolving viscous flows like the unfoced Lamb Oseen votex, the time-dependence may be impotant. The time scale of evolution of the Lamb Oseen votex is T ν = L /ν, wheel is oughly the initial size of the votex (the distance fom the axis whee = max ;seefigue3a). If we take as the elevant time scale fo the gowth of petubations the tansient gowth time scale T max then the assumption of stationaity is valid if T ν /T max 1. Using the time scale fom ou model fo T max, we expect tansient gowth to be much faste than the viscous