Strong echo eect and nonlinear transient growth in. shear ows U.S.A.

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1 Strong echo eect and nonlinear transient growth in shear ows J. Vanneste 1,P. J. Morrison and T. Warn 3 1 Dept. of Phsics, Universit oftoronto, Toronto, Canada. Dept. of Phsics and Institute for Fusion Studies, Universit ofteas at Austin, U.S.A. 3 Dept. of Oceanic and Atmospheric Sciences, McGill Universit, Montreal, Canada. The nonlinear interaction of two disturances ecited successivel in a two-dimensional Couette ow isshown to lead to a transient energ growth. This phenomenon, which is called echo eect and eists in several other phsical sstems, is interesting ecause the energ growth appears long after the energ associated with the original disturances has decaed. Here, the echo eect is studied analticall and numericall in a situation where the nonlinear response has the same order of magnitude as the two ecitations. A sstem of amplitude equations descriing the nonlinear interactions etween three sheared modes is derived and emploed to eamine the phsical mechanism of the echo. The qualitative validit of this sstem is conrmed numerical simulations. The inuence of viscous dissipation on the echo eect is also considered. 1 Introduction It is now widel recognized that the kinetic energ of an innitesimal disturance in a shear ow can e signicantl amplied even if the ow is spectrall stale. Such an amplication, which is then followed a deca in the long-time limit and is generall referred to as transient growth, has originall een identied Orr [14] in a two-dimensional Couette ow. In recent ears, it has een studied in connection with various prolems in hdrodnamics stailit(e.g.[6, 4, 15, 18, ]) and in meteorolog (see the review [7]). The phsical mechanism ehind transient growth is particularl transparent for two-dimensional incompressile ows, which are governed a vorticit equation. In such ows, the disturance vorticit is advected and sheared the asic-ow velocit, and the disturance energ cruciall depends on the phase miing that appears when the streamfunction is derived from the vorticit. In the long-time limit, the scale of the vorticit eld sstematicall decreases with time, leading to an enhanced phase miing and thus to a decrease of the disturance streamfunction and energ [3] this decrease is sometimes referred to as Landau damping, analog with the similar phenomenon of decrease of electrostatic energ in plasmas [1]. However, if the initial disturance is dominated vorticit lines tilted against Present address: DAMTP, Universit of Camridge, Silver Street, Camridge CB3 9EW, UK. 1

2 a E c τ a τ ε t c Figure 1: Schematic representation of the echo eect showing the time evolution of the disturance energ. At t = ;1 a and t =, the shear ow is distured spatial-periodic ecitations. The energ of the response to those ecitations decas through Landau damping, ut ecause of nonlinear eects, a third peak the echo appears in the energ after a time t = ;1 c. the shear, the phase miing does not evolve monotonicall: it temporaril cancels when, after some time, the vorticit lines are perpendicular to the shear, leading to a peak in the disturance streamfunction and energ which is the mark of transient growth. A crucial feature of (linear) transient growth is the fact that it requires ver specic initial conditions: the initial disturances must e coherent, have aver small scale and a specic orientation for the amplication to e signicant [16]. In this paper, we discuss a dierent mechanism that also leads to a transient amplication of the disturance, ut does not require such a small-scale ecitation of the ow. This mechanism, called echo eect, is essentiall nonlinear it is characterized a transient growth of the disturance energ following two successive ecitations, as illustrated schematicall in gure 1. The two successive (spatiall-periodic, impulsive) ecitations, denoted a and, are applied at time t = ;1 a < andt =, respectivel. Through nonlinear interaction, the produce a delaed response, denoted c, the echo. The energ peak associated with the echo is isolated, ecause it appears when the direct responses to the ecitations a and have alread decaed awa through Landau damping. The echo eect is in fact common to a variet ofphsical sstems [9, 1] in particular, it has een studied in plasmas modelled the Vlasov{Poisson equations [8, 13]. In view of the strong analog etween these equations and the equations descriing the evolution of disturances in two-dimensional shear ows, the eistence of an echo eect in shear ows is not entirel surprising. It has in fact een previousl studied Lifschitz [11] who considered a two-dimensional Couette ow inachannel. Following the plasma derivation, he assumed weak amplitudes for the two ecitations and used a regular perturation epansion to calculate the nonlinear response leading to the echo. When this treatment isvalid, the amplitude of the energ peak corresponding to the echo is much smaller than the maimum amplitude of the forced responses a and (although when the echo appears the amplitude of these responses ma have decaed sucientl and e negligile). In this paper, we shall e concerned with a somewhat dierent and more spectacular situation, which we refer to as strong echo and arises when the three energ peaks in the energ have comparale amplitudes. This is possile provided that the amplitude of the two ecitations e O(), with 1, while the time-lag etween them is O( ;1 )( a and c introduced in gure 1 are then O(1) quantities). However, in that case, the perturative treatment of Ref. [11] in not strictl valid we therefore emplo numerical simulations to demonstrate the eistence of a strong echo. We also consideral simplif the prolem dealing with a Couette ow inanunounded domain as opposed to a channel. With this geometr, the disturance vorticit and streamfunction can e epanded in terms of sheared modes (the eact solutions

3 originall discovered Kelvin [17]), whose form is ver simple the echo eect ma theneinterpreted as the nonlinear interaction etween three sheared modes. Eploiting this, we derive a sstem of three amplitude equations (analogous to the three-wave equations for wave triads) which capture the essence of the echo eect. Although these amplitude equations cannot e otained completel rigorousl, the provide results which compare fairl well with numerical simulations. As will e seen, the echo eect involves disturances with small spatial scales, and hence ma e epected to e signicantl aected viscous dissipation. We investigate this inuence deriving an estimate for the viscous damping of the echo and conrm our ndings numericall. Sheared modes in Couette ow The nonlinear evolution of disturances in a two-dimensional Couette ow U = is governed the vorticit @( ) = r! with r =! (1) where is the streamfunction and the inverse of a Renolds numer. This equation has een rendered dimensionless using ;1 as a timescale and a reference length L as a lengthscale. The dimensional viscosit is thus related to through = =(L ). A standard procedure to stud Couette ows is to introduce the convected coordinates (the importance of which was noted e.g. [5] in this prolem) which transform (1) into and give the form X := ; t Y := T @(X Y ) = r! () r =(1+T to the Laplacian. Considering a periodic domain in X and Y,we can epand the vorticit and the streamfunction in Fourier series according to! = = ka=;1 la=;1 A a (T ) ep[i(k a X + l a Y )] (3) ;A a (T ) k a +(l a ; k a T ) ep[i(k ax + l a Y )] where the suscript a of A a designates the pair (k a l a )anda;a =(A a ). Here, we have introduced a formal parameter 1 as we are concerned with weak amplitude disturances. Note that the etension of (3) to an unounded domain is immediatel otained replacing the summations integrals. Introducing (3) into () leads to the amplitude equations da a dt = ; k=;1 l=;1 1 k +(l ; k T ) ; 1 k +(l c c ; k c T ) (k c l ; k l c ) A A c k a +(l a ; k a T ) A a (4) 3

4 where c is dened the interaction conditions k a + k + k c = l a + l + l c =: (5) (We consider sum interactions onl, allowing for oth positive and negative values of the wavenumers.) We now concentrate on a strictl inviscid uid with =. As is well-known, a single mode with constant amplitude A a (T )=A a () is then an eact solution of (4). In the original variales ( t), this solution consists of a sheared mode whose vorticit! = A a () ep fi[k a +(l a ; k a t)]g (6) ehaves like apassive tracer: it is constant along straight lines which are simpl tilted the shear the slope of these lines evolves as 1=(t ; l a =k a ). The time evolution of the disturance kinetic energ RR jr j dd of a sheared mode is given 1 E a (t) = a + l a ja a ()j k +(l a a ; k a t) = k k +(l a a ; k a t) E a(): Although in the long-time limit the energ decas like t ;, it can e temporaril amplied if k a =l a >, i.e. if the vorticit lines are initiall tilted against the shear. It is this transient growth, identied originall Orr [14], that has motivated much of the literature on sheared modes [6, 4,15,18,]. 3 Echo eect A superposition of sheared modes is generall not an eact solution of the nonlinear equation (1). Analzing spectral equations analogous to (4), Tung [19] nevertheless concluded that it represents an O() approimation to an eact solution and is uniforml valid in time, ecause of the eplicit decrease of the nonlinear terms. However, as pointed out Hanes [1] in his stud of the stailit of sheared modes, this conclusion assumes that all the wavenumers l a are O(1). If at least one wavenumer l a is O( ;1 ), i.e. if initiall the disturance partl consists of strongl tilted vorticit lines, the nonlinearit can have non-trivial consequences at leading order which cannot e captured a regular perturation epansion. The echo eect which we now descrie relies on this fact and requires the simultaneous presence of two modes: a mode a with l a = O( ;1 ) and a mode with l = O(1). A natural wa toachieve this conguration is to force the two modes at two successive instants separated a time of O( ;1 ). Successive ecitation have een traditionall considered to stud other echo eects oth theoreticall and eperimentall [9, 1, 8, 13,11] however, in these studies, the time lag is assumed to e much smaller than ;1, leading to an echo amplitude much smaller than the maimum amplitudes of the forced responses. Consider mode a forced at t = ;1 a <, a = O(1), with vorticit lines perpendicular to the shear. The susequent evolution of the vorticit is given (6),with while the energ decas according to l a = ;1 k a a A a () = A a ( ;1 a ) (7) E a (t) = ja a ( ;1 a )j k a[1 + (t ; ;1 a ) ] = 1 1+(t ; ;1 a ) E a( ;1 a ): At t =, when mode a has vorticit lines strongl tilted in the direction of the shear (their slope is ; ;1 a > ) and a ver small energ (E a () E a ( ;1 a )), we ecite the second mode, with k 6= k a 4

5 .5 A a c t Figure : Time evolution of the amplitudes of the three modes a, and c as predicted the truncated sstem (1). and l =. The linear evolution of this mode is again a simple tilt of the vorticit lines, with a deca of the energ. However, the nonlinear interaction of a and generate a third mode c, whose wavenumers satisf (5). Dening, where c is O(1), it can e seen that the vorticit ofc takes the form ;1 c := l c k c = ;1 k a a k a + k (8)! c = A c (t) ep fi[k c +(l c ; k c t)]g = A c (t) ep ik c [ +( ;1 c ; t)] where the amplitude A c (t) is determined the nonlinear interactions. Correspondingl, the energ of c is given ja c (t)j E c (t) = kc[1 + (t ; ;1 c ) ] : If c >, which isachieved if k a a k a + k > i.e. if k a k < and jk a j < jk j (9) one can epect the appearance of a peak in E c for t ;1 c. At that time, oth E a and E have decreased to O( 4 )whilee c is O( ) since, as shown elow, A c ( ;1 c )=O(1). The peak in E c can thus e viewed as the echo of two modes which have alread damped awa (see gure 1). Note that, ecause sheared modes are eact nonlinear solutions of (1) (with = ), we can treat the generation of the echo as an initial-value prolem, assuming that oth disturances a and are initialized at t = with l a given (7) and l =. 5

6 4 Truncated model To investigate the echo eect in more details, we rst consider the truncation of (4) to the triad of modes a c. Neglecting O() terms and using (7) and (8), the evolution equations for A a A A c can e written 8 >< >: da a dt da dt da c dt = ;J 1 k (1 + t ) ; 1 k [1 + (t ; ;1 c c ) ] J = ; k [1+(t ; ;1 c c ) ] A A c a J = k (1 + t ) A A a A A c where J := k k c c = O(1). Clearl, the initial forcing of A c is O(1) and, in general, one can epect the mode c to quickl reach an O(1) amplitude. This can e conrmed direct numerical solutions of (1). As an eample, we consider the modes with k a = ; l a =1 k =3 l = k c = ;1 l c = ;1 which satisf (5) and (9), such that ;1 a = ;5 and ;1 c =1. We take =:1 andchoose the initial amplitudes A a = and A = i. Figure displas the time evolution of the three amplitudes. As epected, mode c attains an amplitude comparale to that of a and due to their nonlinear forcing. For t = O( ;1 ), A c is approimatel constant (ecause of the eplicit time decrease of the nonlinear interaction term), ut A a and A are strongl modulated for t ;1 c = 1. The echo eect appears in the total energ, given E = ja a j k a[1 + (t ; ;1 a ) ] + ja j k (1 + t ) + ja c j kc[1 + (t ; ;1 c ) ] whose evolution is shown in gure 3 (dashed curve). Initiall, the energ is dominated thecontriution of mode and decas as predicted the linear theor (dotted curve). For longer time, however, the contriution of mode c is dominant and the energ ehiits a clear peak for t ;1 c corresponding to the echo. (The energ had also peaked after the ecitation of mode a, att = ;1 a <.) It is possile to derive an approimate analtical solution of (1) and thus to otain an estimate for the echo amplitude noting that with 1 the nonlinear terms are signicant onlfort = O(1) and t ; ;1 c = O(1). For t = O(1), A is almost constant and simpl plas a cataltic role in the interaction etween modes a and c. Similarl, for t ; ;1 c = O(1), A c is almost constant the evolution of A a and during this second time period is irrelevant for the echo eect. Focusing on t = O(1), we approimate A (1) the linear sstem 8 > < >: da a dt da c dt ;J = k (1 + t ) A A c J = k (1 + t ) A A a where A = A () is kept constant. The solution corresponding to the initial conditions A a = A a () and A c =is A a (t) =A a () cos( arctan t) A c (t) = A a()a () sin( arctan t) ja ()j where := ja jj=k. Since A c does not change signicantl for t 1, we otain the following estimates for the echo amplitude: ja c ( ;1 c )jja a () sin(=)j! E echo ja a ()j sin (=): (11) 6 k c (1)

7 .3.5. complete truncated linear E t Figure 3: Time evolution of the total disturance energ in the sstem governed the complete equations (4), in the truncated sstem (1) and according to the linear theor. In the aove eample, ==3, so that the estimates are ja c (( ;1 c )j p 3=1:73 and E echo :3. Both values compare well with what has een found solving (1) numericall, as seen from gures and 3, although is onl marginall small. A few remarks can e made aout the approimate result (11). First, it indicates that the echo amplitude depends on the initial amplitudes of modes a and in two distinct manners: the echo amplitude is directl proportional to A a (), whereas A () merel determines a time scale for the evolution of c. Note also the particular dependence of ja c j on, i.e. on ja j for ed wavenumers (and thus ed J) the echo is maimized for =(n + 1), where n is an integer, and it disappears for =n. Finall, we mention that standard results aout (weak) echo, which neglect the feedack of c on a and, correspond to the limit 1 in (11). This leads to the estimates ja c ( ;1 c )j k c c ja a ()A ()j =:9! k E echo E c( ;1 c ) ja c a ()A ()j 4k =:44: which provide the correct order of magnitude although the are signicantl overestimated as one could epect. 5 Numerical results The sstem (1), which has een derived as an ad hoc truncation of (4), misses important parts of the dnamics of (4), in particular the generation of a mode d,withk d = k ;k a and l d = l ;l a, corresponding to the interaction of a with ;. This and other neglected interactions are likel to weaken the echo. It is therefore important to stud the echo eect directl with the complete sstem (4), or in pratical terms, with a truncation of (4) keeping man modes. It turns out that convergent results are otained with a limited numer of modes those presented elow have een otained for the same parameters as efore, with a 5 5 truncation, and reect the ehaviour of the complete sstem (the disturance energ is 7

8 Figure 4: Disturance streamfunction for t =1:5 4:5 7:5and1:5. Note the change in the contour levels. particularl stale when resolution is varied.) The evolution of the energ so otained is displaed in gure 3 (solid curve). As anticipated, the truncated model signicantl overestimates the energ peak at t ;1 c. Nevertheless, the new results conrm the qualitative validit of the truncated model and the eistence of a strong echo eect. Note that gure 3 also shows the energ evolution predicted the linear theor this theor onl capture the Landau damping of mode and thus completel misses the appearance of the echo. It is interesting to eamine the evolution of the structure of the streamfunction and vorticit during the simulation. Figure 4 shows the streamfunction at t =1:5 4:5 7:5and1:5. At t =1:5, the streamfuction is dominated mode, since Landau damping has alread acted strongl on mode a, while mode c onl egins to emerge. At t = 4:5, the streamfunction amplitudes of modes and c are similar, ut weak ecause oth have relativel small spatial scales. At t = 7:5 and t = 1:5, little remains of mode in terms of the streamfunction which is dominated mode c. The amplitude of the streamfunction increases as the spatial scale of this mode increases it reaches its maimum for t = 1 efore decreasing. Note the factor etween the amplitude of the streamfunction for t =1:5 and for t =1:5 which indicates the importance of the echo eect. It is dicult to plot the evolution of the vorticit eld similarl, since it contains ver small scales associated with mode a, and, after some time, with mode. Figure 5, which shows the vorticit eld with 8

9 Figure 5: Vorticit eld with wavenumer k c for t =7:5 and 1. -wavenumer k c onl, focuses on mode c. Thetwo panels correspond to t =7:5 (efore the echo) and t = 1:5 (after the echo). Although a dominant orientation of the vorticit lines can e distinguished and corresponds to the ehaviour epected from mode c, this gure emphasizes that several sheared modes, with wavenumer k = k c ut dierent wavenumers l, are in fact superposed. As gures 4 and 5 clearl illustrate, the echo eect relies on the presence in the ow of disturances with ver small scales. Since such disturances are strongl aected dissipation, it is important to consider the echo eect with a non-zero viscosit. A rough estimate of the inuence of viscosit can e derived from (4) noting that, in the linear approimation, an mode amplitude is damped a factor ep[;(k + l ; k a a a l a t + k a t =3)t]. The modes a and c involved in the echo have wavenumers l a and l c which are O( ;1 ), and mode c must remain ecited until t ;1 c. Therefore, at the momentoftheecho, dissipation is responsile for an overall damping factor of ep(; ;3 ). A condition for the echo to occur in a viscous uid is thus ;3 1. The validit of this estimate can e conrmed direct numerical resolution of (4) (again with a 5 5 truncation) for dierent values of the viscosit parameter. The corresponding evolution of the total energ is shown in gure 6. With the non-zero values chosen for which are O( 3 ), the energ peak signicantl decreases, although it remains well dened. It is clear that for 3 =:1, the echo would e virtuall unaected dissipation, whereas for 3 it would entirel disappear. 6 Conclusion In this paper, we have studied the echo eect in a two-dimensional Couette ow oth analticall and numericall. This phenomenon can e regarded as a nonlinear mechanism of transient growth: a disturance strongl tilted against the shear, and thus susceptile to eperience a signicant energ amplication, is generated the nonlinear interaction of two other disturances which are ecited successivel. The time lag etween these two ecitations, when large as is assumed here, provides a natural wa of introducing in the sstem disturances with ver dierent scales. This leads to a signicant nonlinear eect although the initial ecitation amplitude is weak. The echo eect, as the instailit of sheared modes [1], thus illustrates the diculties that ma arise when linearizing evolution equations for disturances in stale shear ows: stailit guarantees that some norm of the disturance the enstroph for Couette ows 9

10 E t Figure 6: Time evolution in the sstem governed the complete sstem (4) for dierent values of the viscosit parameter. is ounded, ut this does not preclude the nonlinear terms to e large since the involve vorticit gradients which increase with time. Our investigation of the echo eect is particularl simple, mainl thanks to the ver simple form taken sheared modes in two-dimensional Couette ows when the domain is unounded. The phsical mechanisms involved, however, are generic to all monotonic shear ows, so that echoes can e epected to occur in a variet of situations. Geophsical ows seem especiall interesting in that respect, since the are ver little aected dissipation this can motivate an etension of our work to include the eects of rotation, curvature (-eect) and stratication. The possiilit of spatial echoes [13] also deserves investigations, notal ecause the would e well suited for an eperimental demonstration of the echo eect in shear ows. We thank Prof. A. Lifschitz for calling our attention on his paper. Support from the Isaac Newton Institute for Mathematical Sciences at Camridge Universit, where part of this work was carried out, is gratefull acknowledged. References [1] I.D. Aella, N.A. Kurnit, and S.R. Hartmann. Photon echoes. Phs. Rev., 141:391{46, [] N.J. Balmforth and P.J. Morrison. Singular eigenfunctions for shearing uids. Institute for Fusion Studies Report, no. 69, The Universit of Teas, Austin, Teas, [3] S. Brown and K. Stewartson. On the algeraic deca of disturances in stratied shear ows. J. Fluid Mech., 1:811{816, 198. [4] K.M. Butler and B.F. Farrell. Three-dimensional optimal perturations in viscous shear ow. Phs. Fluids A, 4:1637{165,

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