Stability of the Rankine Vortex in a Multipolar Strain Field

PHYSICS OF FLUIDS VOLUME 13, NUMBER 3 MARCH 2001

Stability of the Rankine vortex in a multipolar strain field Christophe Eloy and Ste´phane Le Dize`s Institut de Recherche sur les Pheńome`nes Hors E´ quilibre, Technopoˆle de Chaˆteau-Gombert, 49 rue F. Joliot Curie, BP 146, 13384 Marseille cedex 13, France ͑Received 7 May 1999; accepted 6 December 2000͒ In this paper, the linear stability of a Rankine vortex in an n-fold multipolar strain field is addressed. The flow geometry is characterized by two parameters: the degree of azimuthal symmetry n which is an integer and the strain strength ␧ which is assumed to be small. For nϭ2, 3 and 4 ͑dipolar, tripolar and quadrupolar strain fields, respectively͒, it is shown that the flow is subject to a three-dimensional instability which can be described by the resonance mechanism of Moore and Saffman ͓Proc. R. Soc. London, Ser. A 346, 413 ͑1975͔͒. In each case, two normal modes ͑Kelvin modes͒, with the azimuthal wave numbers separated by n, resonate and interact with the multipolar strain field when their axial wave numbers and frequencies are identical. The inviscid growth rate of each resonant Kelvin mode combination is computed and compared to the asymptotic values obtained in the large wave numbers limits. The instability is also interpreted as a vorticity stretching mechanism. It is shown that the inviscid growth rate is maximum when the perturbation vorticity is preferentially aligned with the direction of stretching. Viscous effects are also considered for the distinguished scalings: ␯ϭO(␧) for nϭ2 and 3, ␯ϭO(␧2) for nϭ4, where ␯ is the dimensionless viscosity. The instability diagram showing the most unstable mode combination and its growth rate as a function of viscosity is obtained and used to discuss the role of viscosity in the selection process. Interestingly, for nϭ2 in a high viscosity regime, a combination of Kelvin modes of azimuthal wave numbers mϭ0 and mϭ2 is found to be more unstable than the classical helical modes mϭϮ1. For nϭ3 and 4, the azimuthal structure of the most unstable Kelvin mode combination is shown to be strongly dependent on viscous effects. The results are discussed in the context of turbulence and compared to recent observations of vortex filaments. © 2001 American Institute of Physics. ͓DOI: 10.1063/1.1345716͔

I. INTRODUCTION the context of vortex rings.10,11 The mechanism of the instability was explained by Tsai and Widnall12 for an elliptical Recent experiments and numerical simulations demon- Rankine vortex and by Moore and Saffman13 for a general strate the presence of structures of high vorticity in turbu- inviscid vorticity profile. They showed that the instability lence ͑see for instance Refs. 1–3͒. It was argued that those could be interpreted as a resonance phenomenon of the elongated and distorted filaments of vorticity could play an Kelvin modes of the underlying vortex with the elliptic dis- important role in the intermittent character of turbulence.4 tortion. In particular, they established that the combination of Arendt et al.,5 in numerical simulations of gravity waves, stationary helical modes of azimuthal wave numbers m showed that the deformation of those vortex tubes could be ϭϪ ϭ viewed as a superposition of Kelvin modes6 on a straight 1 and m 1 is always resonant and unstable. The filament, but they did not propose a mechanism for the gen- growth of this resonant helical combination leads to a sinu- eration of such perturbations. The aim of this paper is to ous deformation of the vortex in the plane of maximal posi- provide such a mechanism. tive stretching. So far, there has been no proof that this com- This paper focuses on the idealized Rankine vortex flow bination of Kelvin modes is the most unstable, but this is 14–17 which allows a comprehensive analysis. The vortex is sub- supported by numerous experimental observations. In ject to a multipolar potential strain field perpendicular to the this paper, the growth rate of all possible resonant configu- vortex axis which induces a deformation of the streamlines rations will be computed. The conditions under which the in the rotational part of the flow. Our goal is to understand helical mode combination is the most unstable will be estab- under which conditions Kelvin modes can appear spontane- lished. ously on the vortex via a coupling with the multipolar strain. The unstable character of elliptic streamlines was re- In particular, we want to address the role of viscous and examined by Pierrehumbert18 and Bayly19 using a local ap- finite core size effects in the Kelvin mode selection process. proach. They showed that the instability, called elliptic insta- The effect of an external perpendicular strain field on a bility in this framework, could be interpreted as a parametric vortex filament is known to induce, at first order, an elliptic excitation of inertial waves. Lifschitz and Hameiri20 recov- deformation of the streamlines.7–9 The stability of such a ered this result in their general stability theory based on geo- deformed filament was first studied by a global analysis in metrical optics methods. This local theory was also used to

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FIG. 1. Streamlines of the Rankine vortex in a multipolar strain field of strength ␧ϭ0.25. The flow given by Eqs. ͑1.1a͒ and ͑1.1b͒ is pictured at two different scales for dipolar strain field (nϭ2) ͑a,b͒, tripolar strain field (nϭ3) ͑c,d͒ and quadrupolar strain field (nϭ4) ͑e,f͒. The solid and dashed lines are inside and outside the vortex core, respectively.

account for various additional effects such as time depen- 1 ␧ ͑ 21 ␺ ͑r,␪͒ϭϪ r2ϩ rn sin͑n␪͒ϩO͑␧2͒ for rрR, dence, stretching, stratification, etc. see Bayly et al. and in 2 n reference therein͒. The connection between the local and glo- ͑1.1a͒ bal descriptions of the elliptic instability was made by 22 ␧ Waleffe who showed that most unstable inertial waves ␺ ␪͒ϭϪ ͒ϩ Ϫ ͒ nϩ Ϫn ␪͒ out͑r, ln͑r ͓͑n 1 r r ͔sin͑n could be summed up to form the resonant helical Kelvin n2 mode combination of Tsai and Widnall.12 ϩ ͑␧2͒ Ͼ ͑ ͒ Few experimental works have been designed to study the O for r R, 1.1b elliptic instability. Chernous’ko23 studied the flow inside an ␪ ␺ ␪ ϭϪ with R( ) such that in(R, ) 1/2, i.e., elliptic cylinder and gave an instability diagram showing the ␧ number of structures as a function of aspect ratio and eccen- R͑␪͒ϭ1ϩ sin͑n␪͒ϩO͑␧2͒, ͑1.2͒ tricity. Gledzer and Ponomarev15 and Malkus14 confirmed n that these results agree with the Kelvin mode resonance where n is an integer and ␧ is a small parameter measuring mechanism. Experimental observations of the elliptic insta- the strength of the external field. Here, the basic rotation bility have also been evidenced in open flows configurations speed and the vortex radius have been used to nondimension- 10 16 such as vortex rings and vortex pairs. It has also been alize the variables. This solution is the extension of Moore recognized as an important mechanism in the secondary in- and Saffman solution28 to a multipolar strain field in the limit stability transition in shear layers,24,25 wakes17 and rotating of weak external field. Figure 1 pictures the streamlines of flows in confined geometry.26,27 the flow for ␧ϭ0.25 and nϭ2, 3 and 4. For nϭ2, all the In this paper, we perform a global stability analysis of inner streamlines are ellipses with same eccentricity ␧. For the Rankine vortex in a weak stationary multipolar strain larger n, the streamlines are circular in the vicinity of the field. The basic flow is given by the following streamfunc- center and become more deformed close to rϭR(␪). The tions ͓in cylindrical coordinates (r,␪,z)͔: case of pure strain field (nϭ2) was considered by Tsai and

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Widnall.12 Our goal is here twofold. First, we want to recon- A. Governing equations for the perturbations sider the pure strain case to show that other unstable modes Consider a linear perturbation of the basic flow Eqs. different from the helical mode combinations obtained by ͑1.1a͒, ͑1.1b͒ of small amplitude ␭ and assume, as in Tsai Tsai and Widnall exist and that they may become dominant and Widnall,12 that the perturbation is potential outside the if viscosity is included. Second, we want to show that a vortex core. The perturbation is then defined by its velocity similar instability mechanism exists when the pure strain is potential ⌽ outside the vortex core, its four-component replaced by a multipolar strain field which corresponds to a velocity-pressure field vϭ(v ;v␪ ;v ;p) inside the vortex higher symmetrical environment ͑network of vortices, vortex r z ͒ core and the displacement a of the vortex edge. Accordingly in a square box . the core of the perturbed vortex is defined by rрRϩ␭a. 29 we addressed the local stability of In a previous paper, The equations for v are obtained by linearizing the the streamlines defined by Eq. ͑1.1a͒ using the Lifschitz and 20 Navier–Stokes equations around the vortex core expression Hameiri theory. The results can be summarized as follows. ͑ ͒ ϭ ␺ ϭ 1.1a . They can be written in the following condensed form: For n 2, 3 and 4, a given closed streamline in C is un- ץ stable with respect to short wavelength inviscid perturbations JvϩMvϭ␧͑ein␪NϩeϪin␪N¯ ͒vϩ␯LvϩO͑␧2͒, ͑2.1͒ tץ -as soon as ␧Ͼ0 with a leading order growth rate propor tional to the asymmetry parameter ␧ ϵ␧͓2nC/(2 n J M N L Ϫn)͔(nϪ2)/2. For nу5, the streamlines become unstable only where , , and are operators given in Appendix A. ␧ In Eq. ͑2.1͒, N¯ is the matrix whose elements are the complex if they are sufficiently distorted, i.e., if n is above a positive critical value. With the restrictions Ϫ1/2рCр0 and small conjugates of those in N, and ␯ is the kinematic viscosity. ␧, one then expects the flow to be locally unstable for n Note that, with our nondimensionalization, the dimensionless ϭ2, 3 and 4 with a maximum inviscid growth rate given by viscosity ␯ is also the Ekman number of the flow based on ␴ϭ(9/16)␧, (49/32)␧ and 3␧, respectively, and locally the radius of the vortex core. Both parameter ␧ and viscosity stable for larger values of n.29 These results are purely local ␯ are assumed to be small in the following. As seen below and limited to short wavelength perturbations. Therefore, for nϭ2 and 3, the distinguished limit is obtained when both ␯ they are not sufficient to obtain information on the most parameters are of same order. The rescaled viscosity 1 unstable global modes of the vortex. For this purpose, it is ϭ␯/␧ will indeed be the only control parameter in these necessary to use a global theory and to interpret the instabil- cases. The equation for the velocity potential ⌽ outside the ity in terms of Kelvin mode resonance. This constitutes the core is subject of the present paper which is organized as follows. ⌬⌽ϭ0. ͑2.2͒ In Sec. II, it is shown that Kelvin mode resonance is possible only if nϭ2, 3 and 4 and a formal expression of the For small ␧, the equations for the edge displacement a follow growth rate is obtained for nϭ2 and 3. In Sec. III, cases n from the kinematic and dynamic boundary conditions evalu- ϭ2 and 3 are treated quantitatively: the growth rate is com- ated at rϭ1: ⌽ץ ⌽ץ puted for each resonant combination and the role of viscosity ϭ Ϫ ϭ␧ ͑ ␪͒ͩ Ϫ ͪ in the mode selection process is analyzed. Case n 4 is spe- v cos n v␪ ␪ץ rץ r cial, as the resonance occurs for infinite axial wave numbers. 2⌽ץ ץ It requires a separate treatment, which is presented in Sec. ␧ vr IV. In Sec. V a physical explanation of the instability is Ϫ sin͑n␪͒ͩ Ϫ ͪ ϩO͑␧2͒, ͑2.3a͒ 2 ץ rץ n provided in terms of vortex stretching. In particular, it is r aץ aץ shown that the more unstable the mode combinations, the more important the correlation between perturbation vortic- v ϭ ϩ ϩO͑␧͒, ͑2.3b͒ ␪ץ tץ r ity and the direction of stretching of the basic flow. In the ⌽ץ ⌽ ␧ץ ⌽ץ ⌽ץ last section, the results are summarized and discussed in the pϩ ϩ ϭϪ␧ cos͑n␪͒ Ϫ sin͑n␪͒ͩ Ϫn ␪ץ r nץ ␪ץ tץ .context of turbulent flows 2⌽ץ 2⌽ץ pץ ϩ ϩ ϩ ͪ ϩO͑␧2͒. ͑2.3c͒ ␪ץrץ tץrץ rץ II. INSTABILITY MECHANISM AND SCALINGS The system of Eqs. ͑2.1͒–͑2.2͒ with the matching conditions Eqs. ͑2.3a͒–͑2.3c͒ describes the evolution of the linear per- In this section, the calculation of a formal expression for ⌽ the growth rate of the instability is detailed. The presentation turbation (v,a, ) of the vortex in the limit of small strain. follows the papers of Tsai and Widnall12 and Moore and B. Description of the Kelvin modes Saffman.13 However their analysis is extended to account for multipolar strain, general resonant modes and viscous ef- As explained by Tsai and Widnall12 and Moore and fects. The linear equations for the perturbation of the de- Saffman,13 the instability of a deformed vortex patch is the formed Rankine vortex are first given and the Kelvin modes consequence of the coupling of Kelvin modes with the ex- are defined. Then the mechanism of Kelvin mode coupling ternal field. The Kelvin mode perturbations are defined as the with the multipolar strain field is described and analyzed in inviscid normal modes of the underlying axisymmetric basic the limit of weak strain using perturbation theory. flow ͑i.e., the Rankine vortex͒. Thus, they satisfy Eqs. ͑2.1͒,

wave numbers separated by n can be coupled at leading order. The condition of resonance of two Kelvin modes ␻ ␻ Ͼ (k1 ,m1 , 1) and (k2 ,m2 , 2), with m2 m1, is then ϭ Ϫ ϭ ␻ ϭ␻ ͑ ͒ k1 k2 , m2 m1 n, 1 2 . 2.6 This condition of resonance is analyzed in the (k,␻)-plane by looking for the crossing points of the dispersion relations associated with two Kelvin modes of azimuthal wave num- ϭ bers m1 and m2 separated by n. For n 2, this was done by 12 ϭϪ Tsai and Widnall for the two helical modes m1 1, m2 ϭ1. Here property Eq. ͑2.5͒ implies that resonance can occur only if nϭ2or3.Fornϭ4, resonance is also possible for infinite axial wave numbers ͑as branches merge for k→ϱ); ␻ FIG. 2. Dispersion relation of the Kelvin modes in the (k, )-plane for m this case will be studied separately in Sec. IV. For nу5, the ϭϪ1 ͑dashed line͒ and mϭ1 ͑solid line͒. Only the first ten branches are represented. family of branches of two Kelvin modes of azimuthal wave numbers separated by n are totally distinct. Consequently, in such a case, no instability by a Kelvin mode resonance phe- ͑2.2͒, and ͑2.3a͒–͑2.3c͒ with ␧ϭ␯ϭ0. Their velocity- nomenon is possible at leading order. This is in agreement 29 pressure field vK , edge displacement aK and velocity poten- with our previous result that the flow is locally stable to ⌽ у tial K can be written as small n-fold symmetrical deformation if n 5. The end of the present section is dedicated to a formal v ͑k,m,␻͒ϭU͑r͒ei(kzϩm␪Ϫ␻t)ϩc.c., ͑2.4a͒ K calculation of the growth rate of a resonant Kelvin mode ␻͒ϭ␳ i(kzϩm␪Ϫ␻t)ϩ ͑ ͒ ϭ aK͑k,m, e c.c., 2.4b combination when n 2 or 3. The calculation closely follows Moore and Saffman’s presentation.13 We start with a ⌽ ͑ ␻͒ϭ␾͑ ͒ i(kzϩm␪Ϫ␻t)ϩ ͑ ͒ K k,m, r e c.c., 2.4c perturbation (v,a,⌽) of the form

where the axial wave number k, the azimuthal wave number ␪ ␪ ϭ͓ (1)͑ ͒ im1 ϩ (2)͑ ͒ im2 m and the frequency ␻ are connected through a dispersion v A1U r e A2U r e Ϫ␻ ␧␴ relation D(k,m,␻)ϭ0 which is obtained by enforcing con- ϩ␧ ␪͒ i(kz t) 1tϩ ␧2͒ϩ ͑ ͒ v1͑r, ͔e e O͑ c.c., 2.7a ditions ͑2.3a͒ and ͑2.3c͒. The calculation leading to the dis- 30 ␪ ␪ Ϫ␻ ␧␴ ␻ ϭ ϭ ␳(1) im1 ϩ ␳(2) im2 ϩ␧ i(kz t) 1t persion relation D(k,m, ) 0 is classical and will not be a ͓A1 e A2 e F1͔e e reproduced here. Expressions for U(r), ␳, ␾(r) and ϩ ͑␧2͒ϩ ͑ ͒ D(k,m,␻)ϭ0 are given in Appendix A. O c.c., 2.7b It is useful to emphasize a few important properties of ␪ ␪ ⌽ϭ͓A ␾(1)͑r͒eim1 ϩA ␾(2)͑r͒eim2 the dispersion relation. For each azimuthal wave number m 1 2 Ϫ␻ ␧␴ ϩ␧⌽ ␪͒ i(kz t) 1tϩ ␧2͒ϩ ͑ ͒ and axial wave number k, there is a discrete infinity of fre- 1͑r, ͔e e O͑ c.c. 2.7c quencies ␻ satisfying D(k,m,␻)ϭ0. They are all real and in the interval At leading order this perturbation is a combination of two resonant Kelvin modes (k,m ,␻) and (k,m ,␻) with un- mϪ2Ͻ␻Ͻmϩ2. ͑2.5͒ 1 2 known amplitudes A1 and A2. Therefore it is a leading order In Fig. 2 the first branches of the dispersion relation are solution of equations ͑2.1͒, ͑2.2͒ and ͑2.3a͒–͑2.3c͒. The cor- ␧␴ ␻ ϭ ϭϪ ␧ 1t plotted in the (k, )-plane for m 1 and m 1. Note that, rection term v1 as well as the slowly varying factor e when k goes to zero, all the branches accumulate to the fre- are generated by the right-hand side of Eq. ͑2.1͒. They are quency ␻ϭm except a particular branch which tends to ␻ due to the interaction of Kelvin modes with the nonaxisym- ϭmϪsgn(m). This property is true for all mÞ0. For m metric part of the basic flow and to the viscous damping of ϭ0, this particular branch does not exist. the perturbation. The frequency ␻ being real, the temporal growth rate of ␴ϭ␧ ␴ C. Kelvin mode resonance and instability the resonant Kelvin modes is then given by Re( 1). In Tsai and Widnall,12 ␴ is obtained by fully integrating Eqs. The Kelvin modes are neutral. The Rankine vortex with- ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ␧ out external field is then expected to be marginally stable to 2.1 , 2.2 and 2.3a – 2.3c up to the order . Here we ͑ prefer to follow the simpler calculation of Moore and vortex core perturbations. A convincing proof of this state- 13 Saffman’s which is based on a solvability condition for v1. ment is still lacking contrarily to what is claimed in Arendt ͑ ͒ et al.31͒ However, as explained by Moore and Saffman,13 a The form of the right-hand side of Eq. 2.1 indicates that v1 small asymmetry is sufficient to couple Kelvin modes and can be written as ␪ ␪ Ϫ ␪ ϩ ␪ may lead to exponential growth. The coupling term associ- ϭ im1 ϩ im2 ϩ i(m1 n) ϩ i(m2 n) v1 V1e V2e V3e V4e . ated with the multipolar strain field is the first term on the ͑2.8͒ right-hand side of Eq. ͑2.1͒. The azimuthal dependence eϮin␪ ͑ ͒ of this term implies that only Kelvin modes of azimuthal Inserting this expression in Eq. 2.1 gives for V1 and V2

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͑Ϫi␻JϩM ͒V ϭϪA ␴ J U(1)ϩA ␯ L U(1) m1 1 1 1 1 1 m1

ϩA N¯ U(2), ͑2.9a͒ 2 m2 ͑Ϫi␻JϩM ͒V ϭϪA ␴ J U(2)ϩA ␯ L U(2) m2 2 2 1 2 1 m2 ϩA N U(1), ͑2.9b͒ 1 m1 where the notation M means M m1 m1 ␪ϩ Ϫ ␪ϩ ϭ͓Mei(m1 kz)͔e i(m1 kz). The integration of these equations is not needed. Introducing the following scalar product:

1 ͗ ͉ ͘ϭ ͵ ͑¯ ϩ¯ ϩ¯ ϩ¯ ͒ ͑ ͒ X Y XrY r X␪Y ␪ XzY z X pY p rdr, 2.10 0 FIG. 3. Dispersion relation of the Kelvin modes in the (k,␻)-plane for ϭ ͑ ͒ ϭ ͑ ͒ m1 10 solid line and m2 13 dashed line . The numbers label the dif- the solvability condition is obtained by forming the scalar ferent branches as explained in the text. The growth rates of the modes product of U(1) and U(2) with Eqs. ͑2.9a͒ and ͑2.9b͒, respec- corresponding to the different intersection points are displayed in Table II. tively. It leads to coupled amplitude equations for A1 and A2: ␴ J ϩ ͒Ϫ␯ L ␴ J ϩ ͒Ϫ␯ L ϩ ␴ ϭϪ ␴ J ϩ ␯ L ϩ N¯ ͓ 1͑ 1͉1 J1 1 1͉1͔͓ 1͑ 2͉2 J2 1 2͉2͔ A2I1 A1 1J1 A1 1 1͉1 A1 1 1͉1 A2 1͉2 , ͑ ͒ 2.11a ϭ N¯ Ϫ ͒ N Ϫ ͒ ͑ ͒ ͑ 1͉2 I1 ͑ 2͉1 I2 . 2.14 ϩ ␴ ϭϪ ␴ J ϩ ␯ L ϩ N A1I2 A2 1J2 A2 1 2͉2 A2 1 2͉2 A1 2͉1 , This relation is the formal expression of the growth rate we ͑ ͒ 2.11b were looking for. For a given resonant Kelvin mode combi- ␴ where the following notation has been used: nation, the growth rate 1 only depends on the rescaled vis- ␯ cosity 1. Here, the difficulty is that the number of resonant N ϭ͗ (2)͉N (1)͘ ͑ ͒ 2͉1 U U . 2.12 Kelvin mode combination is doubly infinite. For each n, there is an infinity of possible couples of azimuthal wave The coefficients I , I , J and J are boundary terms de- 1 2 1 2 numbers (m ,m ) which satisfy the relation m Ϫm ϭn. fined in rϭ1by 1 2 2 1 And for a given couple (m1 ,m2), there exists as mentioned (2) ␻ .2␾(2) above infinitely many crossing points in the (k, )-planeץ U iץ i 1 ϭ (1)ͩ (2)Ϫ r Ϫ ␾(2)ϩ ͪ ͑ ͒ ͑ ͒ I P U␪ im However the dispersion relation Eq. A13 and Eq. 2.14 are 2 2 ץ 1 r ␻ץ n r n 2 invariant by the following symmetry: ( ,k,m1 ,m2) ␾(2) →(Ϫ␻,k,Ϫm ,Ϫm ). Therefore, the study can readily beץ ␻Ϫ ϩ (2) ץ 1 i P m2 n 1 2 Ϫ U(1)ͩ ϩm ␾(2)ϩ ͪ , restricted to mode combinations satisfying m ϩm у0. r 1 2ץ r 2 nץ r n 2 In the next section, the different coefficients of Eq. ͑2.13a͒ ͑2.14͒ are computed for a large range of resonant Kelvin mode combination using the definitions of the operators (2␾(1ץ U(1) iץ i 1 ϭ (2)ͩ (1)ϩ r Ϫ ␾(1)Ϫ ͪ given in Appendix A. In Appendix B, these coefficients are I2 P U␪ im1 .r2 analytically evaluated in the limit of large k and large mץ r nץ n 2 (␾(1ץ ␻Ϫ Ϫ (1) ץ 1 i P m1 n III. DIPOLAR AND TRIPOLAR STRAIN FIELD nÄ2,3 ϩ U(2)ͩ ϩm ␾(1)ϩ ͪ , „ … rץ r 1 nץ r n 2 This section focuses on the cases nϭ2 and nϭ3, for ͑2.13b͒ which the formal expression ͑2.14͒ for the growth rate ap- plies. In a first subsection, the viscous damping is neglected J ϭϪU(1)␾(1), ͑2.13c͒ ␯ ϭ 1 r which amounts to considering 1 0. This permits to com- pare our expression with the previous results of Tsai and ϭϪ (2)␾(2) ͑ ͒ 12 J2 Ur , 2.13d Widnall and with the local and inviscid results obtained in Ref. 29. In a second subsection, the effect of viscosity is where U(1) , U(2) and P(1), P(2) are the radial velocity and r r analyzed and the growth rate of the most unstable Kelvin pressure of the modes U(1) and U(2). These boundary terms mode combinations is computed as a function of ␯ . come from the O(␧) correction terms in the matching con- 1 ditions Eqs. ͑2.3a͒–͑2.3c͒. Note that the external potential A. Inviscid analysis flow intervenes in the amplitude equations ͑2.11a͒,͑2.11b͒ ␯ ϭ ͑ ͒ via the boundary terms I1 , I2 , J1 and J2 and the dispersion When 1 0, Eq. 2.14 reduces to a simple expression relation only. for the inviscid growth rate: ␴ The equation for the growth rate 1 is obtained by re- ͑N¯ ͉ ϪI ͒͑N ͉ ϪI ͒ quiring that the determinant of the linear system Eqs. ␴2ϭ 1 2 1 2 1 2 ␧2 ͑ ͒ i ͑J ϩ ͒͑J ϩ ͒ . 3.1 ͑2.11a͒,͑2.11b͒ vanishes: 1͉1 J1 2͉2 J2

␴ ␧ ϭϪ TABLE I. Table of the computed inviscid growth rate i / for m1 1, TABLE III. Table of the axial wave numbers k of the most unstable modes ϭ р m2 1 as a function of the branch label. The columns are the different (m1 ,m2 ,i) in the dipolar and tripolar cases, for i 4. branches of the dispersion relation for m1 and the lines for m2. The under- lined growth rates are for the principal modes (Ϫ1,1,2), (Ϫ1,1,3), etc. Dipolar Tripolar Ϫ Ϫ 12345 i ( 1,1,i) (0,2,i)(1,2,i) (0,3,i) (1,4,i)

1 – 0.0041 0.0046 0.0038 0.0031 1 0.0000 1.2422 2.2920 3.9360 5.3810 2 0.0041 0.5708 0.0073 0.0061 0.0051 2 2.5050 3.3701 6.1719 7.8218 9.3722 3 0.0046 0.0073 0.5695 0.0061 0.0053 3 4.3491 5.2264 9.7861 11.471 13.078 4 0.0038 0.0061 0.0061 0.5681 0.0051 4 6.1740 7.0584 13.371 15.076 16.718 5 0.0031 0.0051 0.0053 0.0051 0.5672

4͑b͔͒ are qualitatively different. For nϭ2, the inviscid growth rate is maximum for the combination (Ϫ1,1,2). It For each m, we choose to label by an integer each branch of asymptotes the limit value (9/16)␧ when k→ϱ͓dash-dotted the dispersion relation in the order of increasing k ͑see Fig. lines in Fig. 4͑a͔͒. For nϭ3, the growth rate is a decreasing 3͒. We computed expression ͑3.1͒ for 10 000 resonant Kelvin function of k for fixed m and reaches a limit value mode combinations ͑the combinations corresponding to the 1 49␧/(8␲2)Ϸ0.62␧ as k→ϱ͓lower dash-dotted lines in Fig. 100 crossing points of the first 10 branches of the dispersion 4͑b͔͒. As can be noticed in Fig. 4͑b͒, this limit value is much relations of the azimuthal wave numbers m and m ϭm 1 2 1 smaller than the maximum growth rate (49/32)␧Ϸ1.53␧ ϩn with nϭ2 and 3 and Ϫ1рm р48). As an illustration, 1 which is attained when m →ϱ for fixed iÞ1 ͑upper dash- typical values of the growth rate ␴ /␧ are given for the mode 1 i dotted line͒. combinations (m ,m )ϭ(Ϫ1,1) and (m ,m )ϭ(10,13) in 1 2 1 2 In Sec. V, a physical interpretation will be given ex- Tables I and II ͑for the crossing points displayed in Figs. 2 plaining why some modes have larger growth rates than oth- and 3͒. ers. The asymptotic analysis for large k and large m1 that In this computation, we observed that all resonant ϭ ͑ ␴ provides the limit values for n 2 and 3 are detailed in Ap- Kelvin mode combinations are unstable i.e., i is always ͒ р real . We also found that, for m1 48, growth rates are significantly larger for the crossing points of branches with the same label. In the following, these combinations will be named principal modes and numbered using the notation ͒ ͑ ͒ ͑m1 ,m2 ,i , 3.2 ϭ where i is the common label for m1 and m2. For n 2, m1 ϭϪ ϭ 1, m2 1, these crossing points are exactly on the axis of symmetry ␻ϭ0 of the branch family ͑see Fig. 2͒. They are the resonant stationary helical modes studied by Tsai and Widnall12 and Moore and Saffman.13 The principal mode (Ϫ1,1,1) of vanishing axial wave number correspond to a global translation of the vortex. As in Tsai and Widnall, it will be discarded in the following. For all the other cases, principal modes do not exhibit particular symmetries. Note that the first principal mode (m1 ,m2,1) always involves the ␻ ϭ particular m2 branch issued from the point (k, ) (0,m2 Ϫ1). In Table III the axial wave numbers k of the first princi- ϩ pal modes (m1 ,m1 n,i) are given. For given i and n, this ϱ axial wave number increases with m1 and tends to as m1 →ϱ. The variations of the principal mode growth rate as a function of the axial wave number k are displayed in Fig. 4. Note that the graphs for nϭ2 ͓Fig. 4͑a͔͒ and nϭ3 ͓Fig.

ϭ ϭ TABLE II. Same as Table I for m1 10, m2 13. The boxed-in growth rate corresponds to the combination examined in Fig. 12.

␴ ␧ ϩ FIG. 4. Inviscid growth rates i / for the principal modes (m1 ,m1 n,i)as a function of their axial wave number k for dipolar strain ﬁeld (nϭ2) ͑a͒ ϭ ͑ ͒ ϩ and tripolar strain ﬁeld (n 3) b . In both cases, the symbols are: : m1 ϭϪ ᮀ ϭ छ ϭ ᭝ ϭ ᭺ ϭ 1, : m1 0, : m1 1, : m1 10, : m1 20 and the solid and the ϩ dashed lines are the inviscid growth rate of the modes (m1 ,m1 n,1) and ϩ Ϫ р р (m1 ,m1 n,2), respectively, for 1 m1 48. The dash-dotted lines are ␴ ϭ ␧ ͑ ͒ ␴ asymptotic values for large wave numbers: i (9/16) in a ; i ϭ ␧ ␲2 ␴ ϭ ␧ ͑ ͒ 49 /(8 ) and i (49/32) in b .

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pendix B. The main results of this analysis are now summarized. In the limit of large k, for nϭ2, the asymptotic inviscid growth rate is proved to be maximum for principal modes. For other resonant combination, the growth rate tends to zero for large k. For nϭ3, the large k asymptotic growth rate is given by

49␧ ␴ ϭ ͑ ͒ i , 3.3 8␲2͑2iϪ2 jϪ1͒2

where i and j are the branch labels for m1 and m2, respectively. This asymptotic growth rate is maximum not only for the principal modes (iϭ j) but also for the adjacent modes ␻ defined by the condition iϭ jϩ1. The asymptotic results for FIG. 5. Four components of the Kelvin mode vK(k,m, ) ϭ i(kzϩm␪Ϫ␻t) ϭ ␻Ӎ Ӎ ͑ large m are different. For both nϭ2 and nϭ3, the principal U(r)e for m 39, 40.5 and k 51.9 corresponding to a 1 resonance with mϭ42). We have Uϭ(iU;V;W;P) with U in the solid line, modes are no longer the most unstable modes. The growth V in the dotted line, W in the dashed line and P in the dash-dotted line. rate of the first principal modes (m1 ,m2,1) even goes to zero as m1 goes to infinity. The large m1 asymptotic growth rate is now maximum for adjacent modes iϭ jϩ1 with jÞ1. For nϭ2 this maximum is the same as the maximum large k asymptotic growth rate (9/16)␧. The situation is different for vortex has negligible influence on the stability results in the ϭ ␴ n 3. The maximum large m asymptotic growth rate i limit of large wave numbers. Ϸ1.53␧ is much larger than the maximum large k asymptotic ␴ Ϸ ␧ growth rate i 0.62 . B. Viscous selection process The results for m ϭϪ1 and m ϭ1 ͑see Fig. 2 and 1 2 In the previous section the maximum inviscid growth Table I͒ can directly be compared to those of Tsai and rate for nϭ3 was shown to asymptote its maximum value in Widnall.12 The inviscid growth rates and axial wave numbers the limit of a large azimuthal wave number. This is no longer computed here are in agreement with their results for the principal modes (Ϫ1,1,2) and (Ϫ1,1,3). However, we expected to hold when viscosity is included, as viscous ef- found that all Kelvin mode combinations are unstable fects damp perturbations with large wave numbers. More- ϭ whereas Tsai and Widnall12 found that some may be stable. over, we shall see below that, for n 2, another resonant Ϫ A conclusion similar to ours has been reached by Vladimirov combination than the principal mode ( 1,1,2) may become and Il’in32 for the Kirchhoff vortex which makes us confi- the most unstable in the viscous regime. dent of our result. Note, however, that this discrepancy is not In the present paper, only volumic viscous effects on the expected to strongly affect Tsai and Widnall’s conclusions as perturbation are considered. In particular, we neglect the vis- the resonant combination they found stable are in fact ampli- cous diffusion of the basic flow as commonly done in stabil- fied with a negligible growth rate ͑50 times smaller than the ity analysis of inviscid solutions.33 The effect of basic flow maximum growth rate͒. diffusion has been analyzed elsewhere.34 In addition, the vis- The large wave number growth rates (9/16)␧ (nϭ2) cous effects on the perturbation due to the boundary layer at and (49/32)␧ (nϭ3) correspond to the local maximum the edge of the vortex core are not considered. These growth rates of the most unstable streamline in the vortex effects35 are O(␯1/2), but they are not strongly dependent on 29 core. This result was not guaranteed. Indeed, the dispersion the perturbation wave number and therefore are not expected relation for the Kelvin modes is a constraint on the axial and to influence the inviscid selection process. Moreover, these azimuthal wave numbers which is not present in the local viscous layers are not present for more realistic vortices with stability analysis. Note, however, that the value of the local continuous vorticity profile. It is then natural not to take into maximum growth rates are reached in the limit of large azi- account these effects if one wants to obtain generic instabil- muthal and axial wave numbers for nϭ3 while only large ity scenari. Note finally that both boundary layer effects and axial wave number is needed for nϭ2. This could be related viscous diffusion effects are negligible if kӷ␯Ϫ1/4. The as- to the differences in the local stability properties for nϭ2 sumptions made here are therefore fully justified in the limit and nϭ3. For nϭ2 the local stability properties are the same for all streamlines in the vortex core while for nϭ3 the of large k. ͑ ͒ flow is not uniform and the most unstable streamline is lo- In the growth rate expression Eq. 2.14 , volumic viscos- ␯ L ␯ L cated near the boundary where the strain rate is maximum. ity effects appear via the terms 1 1͉1 and 1 2͉2. Each of Thus, for nϭ3, resonant Kelvin mode combinations have to these terms can roughly be decomposed into three parts: one 2 2 be localized near the boundary to be the most unstable and proportional to k , another proportional to m1 and a third one this occurs only for large m1 as illustrated in Fig. 5. Note mostly connected to the geometrical character of the Kelvin Ϫ␯ 2 finally that the agreement with the local stability properties modes. Thus, the viscous damping is of the form 1C1k Ϫ␯ 2ϩ 2 2 of the vortex core shows that the potential flow outside the 1C2m1 o(m1 ,k ) where C1 and C2 are two positive

␯ ϭ changes with 1. For n 2, the most unstable Kelvin mode combination is (Ϫ1,1,2) for small viscosity and (0,2,1) for larger viscosity. This was expected from inviscid analysis since, among all other modes, mode (Ϫ1,1,2) has the largest inviscid growth rate and mode (0,2,1) has the smallest axial wave number. For nϭ3, the most unstable Kelvin mode combination is always a principal mode of the form ϩ ␯ (m1 ,m1 3,1) with m1 increasing for decreasing 1. Again, ␯ → the inviscid results are recovered when 1 0: the most unstable combination corresponds to modes with the infinite azimuthal wave number m1. ϭ ␯ ␯ For both n 2 and 3, there is a viscous cutoff 1c ( 1c Ӎ ϭ ␯ Ӎ ϭ 0.111 for n 2 and 1c 0.122 for n 3) above which all Kelvin mode combinations are damped. For a dipolar strain field, this result can be compared to those obtained by Land- man and Saffman.36 Using a local approach, they found that there exists a critical Ekman number ͑here equal to the dimensionless viscosity ␯), depending on the wave number of the perturbation and on the strain strength, above which the elliptic flow is stable. In the limit of small strain, Landman ␯ Ӎ 2 and Saffman’s threshold reads 1c 0.131/k which becomes ͑ ͒ ␯ for the smallest resonant wave number see Table III 1c Ӎ0.084. This estimate is within 25% from our value. Note, however, that Landman and Saffman’s approach is unable to give estimates of the viscous damping for modes with a more complex azimuthal structure. In addition to the difference in the selected modes, one can also notice that there is a much ϩ ϭ FIG. 6. Normalized growth rate of the principal modes (m1 ,m1 n,i)asa larger variation of inviscid growth rates in case n 3 than in ␯ ͑ ͒ ϭ ͑ ͒ ϭ function of 1. a Dipolar strain (n 2) and b tripolar strain (n 3). Solid case nϭ2. The mode selection is then expected to be less ϭϪ ϭ ϭ lines: m1 1, dashed lines: m1 0 and dotted lines: m1 1. Here, for the efficient for nϭ2 and for small viscosity. sake of legibility, only the first nine principal modes are plotted but the The structure of several resonant Kelvin mode combina- tendency is unchanged for higher modes (iϾ3, m Ͼ1). 1 tion is shown in Figs. 7͑a͒,7͑b͒. The deformation of the vortex core by the perturbation has been depicted using Eq. ͑ ͒ ϭ Ϫ slowly varying functions of k and m1. For a given viscosity, 2.3b . For n 2, the combination ( 1,1,2) gives rise to an the viscous growth rate is then maximum for finite k and m1. undulation of the vortex in the plane of maximal stretching In other words, viscosity selects a particular resonant Kelvin ͓Fig. 7͑a͔͒. This is in agreement with experimental mode combination. observations.14,16,23 For larger viscosity, the principal mode The result of this selection process for nϭ2 and nϭ3is (0,2,1) becomes the most unstable combination. The growth ϭ illustrated in Fig. 6. In these figures, the viscous growth rate of this mode implies the bulging (m1 0) and splitting (m2 ␯ ϭ ͓ ͑ ͔͒ is computed as a function of 1 for the first principal modes 2) of the vortex core see Fig. 7 b . This result is new and Ϫ р р р р 12 (m1 ,m2 ,i) with 1 m1 1 and 1 i 3. The principal shows the limitation of the analysis of Tsai and Widnall modes are still the most unstable wave combinations when which only considered helical mode resonance. It will be viscosity is added but it can be seen in Fig. 6 that the relative discussed again in light of a recent numerical simulation37 in stability of one combination with respect to the others Sec. VI. Note finally that for very small viscosity, the growth

FIG. 7. Illustration of the Rankine vortex deformation induced by the principal modes (Ϫ1,1,2) ͑a͒ and (0,2,1) ͑b͒. The vortex core displacement is represented for perturbation amplitude equal to ␭ϭ0.1 and 0.2, re- ͑ ϭ spectively with A1 1).

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FIG. 8. Same as Fig. 7 for the principal modes (Ϫ1,2,1) ͑a͒, (0,3,1) ͑b͒ and (1,4,1) ͑c͒ with ␭ϭ0.2, 0.1 and 0.1, respectively.

rate of most principal modes becomes comparable ͑within Following the analysis of Sec. II, the perturbation is then ␯ Ͻ Ϫ4 2% for 1 10 ). Depending on the initial conditions, one taken in the form could then imagine that modes with a more complex azi- ␪ (1) ␪ Ϫ (2) ϭ͓ ͑ ͒ (1)͑ ͒ im1 if Tϩ ͑ ͒ (2)͑ ͒ im2 if T͔ muthal structure could also develop. This would give a v A1 T U r e e A2 T U r e e simple explanation of the rich dynamics of vortex filaments ϫ i(kzϪ␻t)ϩ␧ ␪ ͒ ͑ ͒ e v1͑r, ,z,t , 4.4 observed in large Reynolds number flows. For nϭ3, the structure of the most unstable combination where A1(T) and A2(T) are the amplitudes which vary on ␯ the slow time scale Tϭ␧t, of the resonant Kelvin modes strongly depends on viscosity. Close to the threshold 1c , ͑ ͒ ␯ v (k,m ,␻Ϫ␧ f (1)) and v (k,m ϩ4,␻ϩ␧ f (2)). We shall the most unstable mode is as illustrated in Fig. 8 a . When 1 K 1 K 1 is progressively decreased, the most unstable combination see below that, contrary to Sec. II, functions A1(T) and becomes more and more complex. It first gains a three-strand A2(T) are not necessarily exponentials of same argument. ␻ ͑ ͒ structure ͓Fig. 8͑b͔͒, then a four-strand structure ͓Fig. 8͑c͔͒, Expressions for vK(k,m, ) are given in Eq. A10 . Note Ϸ ⌽Ϸ and so on. that, in this limit, a 0 and 0 such that the mode is localized in the core of the vortex. The frequency offsets ⌬␻ϭϪ␧ f (1) and ⌬␻ϭ␧ f (2) depend, via relation ͑A11͒,on ␦ ␦ IV. QUADRUPOLAR STRAIN FIELD nÄ4 1 and 2 which are obtained through the dispersion relation „ … ͑A13͒: ϭ In Secs. II and III, we considered the dipolar (n 2) and Ϫ␦ Ј ␦ ͒ϩ ␦ ͒ϭ ͑ ͒ 1J͉ ͉͑ 1 m1J͉m ͉͑ 1 0, 4.5a tripolar (nϭ3) strain fields for which there exist Kelvin m1 1 mode resonances at finite axial wave numbers k. For the ␦ Ј ␦ ͒ϩ ␦ ͒ϭ ͑ ͒ 2J͉ ͉͑ 2 m2J͉m ͉͑ 2 0. 4.5b quadripolar strain field (nϭ4) such resonances do not exist, m2 2 Ϫ Ͻ␻Ͻ ϩ ␻ since m 2 m 2 for finite k. However, (k,m1 , ) and These implicit equations are solved numerically. Again, each ϩ ␻ ␻ϭ ϩ (k,m1 4, ) may resonate for infinite k if m1 2. In equation has an infinite number of solutions which corre- such a case, the resonant state is singular and the analysis of spond to the different branches of the dispersion relation and Sec. II does not apply. To treat this large k resonance, a which can be numbered by the branch label. The amplitude specific asymptotic analysis is carried out in this section. equations for A1 and A2 can be obtained by the same proce- If the axial wave number k is large but not infinite, the dure as in Sec. II. They now have the form resonance is imperfect and there is an offset of frequency for Aץ each Kelvin mode, which is deduced from Eq. ͑A11͒: 1 i2 fT ϪJ ͉ ϩ␯ L ͉ A ϩN¯ ͉ e A ϭ0, ͑4.6a͒ T 1 1 1 1 1 2 2ץ 1 1 ͉⌬␻͉ϭ␦2/k2. ͑4.1͒ Aץ The distinguished limit is obtained when this offset is of ϪJ 2 ϩ␯ L ϩN Ϫi2 fT ϭ ͑ ͒ 2͉2A2 2͉1e A1 0, 4.6b 1 ץ 2͉2 same order as the O(␧) inviscid growth rate and the viscous T damping rate. This corresponds to the scalings: with 2 f ϭ f (1)ϩ f (2). The large k assumption allows the vis- ␦ cous terms to be reduced to their leading order expression: ͱ 2 2 2 ϭO͑ ␧͒, ␯k ϭO͑␧͒. ͑4.2͒ ␯ L ͉ ϭϪ␯ k J ͉ and ␯ L ͉ ϭϪ␯ k J ͉ . This k 1 1 1 2 Ϫ1/2 1 1 1 2 2 2 Ϫ1/2 2 2 permits one to obtain simple expressions for A1 and A2: It is then useful to rescale the viscosity and axial wave num- ͑ ͒ϭ ͓͑ ϩͱ␴2Ϫ 2Ϫ␯ 2 ͒ ͔ ͑ ͒ ber as follows: A1 T c1 exp if 0 f 2kϪ1/2 T , 4.7a 2 2 2 ␯ A ͑T͒ϭc exp͓͑Ϫifϩͱ␴ Ϫ f Ϫ␯ kϪ ͒T͔, ͑4.7b͒ ␯ ϭ ϭ ͱ␧ ͑ ͒ 2 2 0 2 1/2 2 , kϪ1/2 k . 4.3 ␧2 with

FIG. 9. Stability of the Rankine vortex in a quadrupolar strain ﬁeld. Maxi- ␴ ϩ mum inviscid growth rate 0 of the principal modes (m1 ,m1 4,1) as a function of the azimuthal wave number m1. The dashed line is the limit ␴ ϭ value for large m1 : 0 3.

N c2 2͉1 ϭ , ͑4.8͒ c J ͑Ϫ ϩͱ␴2Ϫ 2͒ 1 2͉2 if 0 f N N¯ ␴ ϭ 2͉1 1͉2 ͑ ͒ 0 J J . 4.9 1͉1 2͉2 The growth rate of the resonant Kelvin mode combination then reads ␴ ϭͱ␴2Ϫ 2Ϫ␯ 2 ͑ ͒ ␧ 0 f 2kϪ1/2 , 4.10 FIG. 10. Viscous effects on the instability characteristics of the Rankine ␴ ͑ ͒ vortex in a quadrupolar strain field. Growth rate max a and corresponding max ͑ ͒ ϩ where f is a O(1) function which depends on the labeled axial wave number kϪ1/2 b of the principal modes (m1 ,m1 4,1) as a ␯ ␦ ␦ ͑ ͒ ͑ ͒ function of 2. Each dotted line stands for a different azimuthal wave num- solutions 1(i) and 2( j) of Eqs. 4.5a , 4.5b : ͑ ͒ ͑ ͒ ber as labeled on a .On b , m1 is increasing from bottom to top. The solid ␦2͑i͒ϩ␦2͑ j͒ line is the most unstable mode after a maximization over m1. ϭ 1 2 ͑ ͒ f 2 . 4.11 2kϪ1/2 Ӎ ϭϪ Ϫ ͒ The first term in Eq. ͑4.10͒ is associated with inviscid effects 89, 56 and 1357 for m1 2, 1 and 0, respectively . Ͼ only. Contrary to nϭ2 and nϭ3, there is a frequency cutoff: For kϪ1/2 kc , the maximum growth rate is always smaller 2у␴2 ϭ if f 0, the inviscid growth rate vanishes. This condition than the growth rate of the first principal mode for m1 1. can be interpreted as a condition of resonance. Indeed, f mea- Consequently the most unstable configuration is always a sures the renormalized gap between the two Kelvin mode first principal mode of the form (m1 ,m2,1). Viscosity does frequencies. If f is too large, there is no resonance anymore. not modify this conclusion. However, since it damps large Moreover, the inviscid growth rate is a decreasing function wave number perturbations, it plays an important role in the of f. As expected, it is maximum for a perfect resonance ( f wave number selection. ␯ ϭ0) which occurs when kϭϱ. For fixed m1 and 2, there exists an optimal wave num- max ␴ϭ␴ To determine the maximum growth rate, one has to com- ber kϪ1/2 which maximizes the growth rate max . Both ␦ ␦ ͑ ͒ ͑ ͒ ␴ max ␯ pute 1(i) and 2( j) from Eqs. 4.5a , 4.5b and the scalar max and kϪ1/2 are drawn in dotted lines as a function of 2 ͑ ͒ ͑ ͒ ͑ ͒ products involved in Eq. 4.9 . This computation was carried for different values of m1 in Figs. 10 a and 10 b , respec- Ϫ р р р ͑ ͒ out for the range of parameters 2 m1 170, i 10 and tively. The solid line in Fig. 10 a is the maximum growth р j 10. The results can be summarized as follows. When rate among all possible m1. The corresponding axial wave у ␴ ͑ ͒ m1 1, 0 is maximum for the principal mode (m1 ,m2,1). number is the solid line in Fig. 10 b . Despite the scaling ␴ ͑ ͒ ͑ ͒ Figure 9 shows the evolution of 0 for this mode as a func- difference, Fig. 10 a exhibits the same features as Fig. 6 b ϭ tion of m1. Since f is an increasing function of the labels, for n 3: When viscosity is decreased below a critical value ␯ Ӎ ϫ Ϫ4 other resonant Kelvin modes necessarily have a smaller 2c 7.5 10 , the combination of modes with the smallest у ͑ ϭϪ ϭ growth rate. For m1 1, then, the first principal mode is azimuthal wave number here m1 2, m2 2) is first de- always the most unstable combination whatever kϪ1/2 and stabilized; then, combinations with higher m1 progressively ␯ ϭϪ Ϫ ␯ → 2. When m1 2, 1 or 0, the first principal mode is the become the most unstable as 2 0. Note, however, that the Ϫ most unstable only if kϪ1/2 is below a critical value kc (kc principal modes ( 1,3,1) and (0,4,1) are never the most

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␯ unstable for any 2. Note also that the first destabilized com- Outside the vortex core, both basic flow and Kelvin ϭϪ ϭ bination m1 2, m2 2 is stationary by symmetry as for modes have zero vorticity so no amplification can be gener- nϭ2. ated in this potential region. However, in the vortex core, As seen in Figs. 10͑a͒ and 10͑b͒, the most unstable com- Kelvin modes exhibit vorticity components in the (x,y) ␯ → ͑ ͒ bination when 2 0 has both a large kϪ1/2 and a large m1. plane in Cartesian coordinates where the stretching term A simplified expression of the growth rate can be obtained ͓third term in Eq. ͑5.1͔͒ acts. Therefore these vorticity com- ͑ ͒ by asymptotic methods as in Appendix B 2 in the large m1 ponents can be amplified by stretching when they are aligned limit: with the direction of stretching. In order to study this possi- bility, we shall now evaluate the correlation between the di- ␴ 4 m1 rection of stretching and the direction of the vorticity pro- ϭͱ9Ϫ Ϫ␯ k2 . ͑4.12͒ ␧ 4 2 Ϫ1/2 jected onto the (x,y) plane for a given Kelvin mode and for kϪ1/2 a combination of two resonant Kelvin modes. ␯ ϭ →ϱ ␴ ϭ For 2 0 and kϪ1/2 , the inviscid growth rate i 3of 29 The local stretching rate s(r) and the local direction of the local stability analysis is recovered. ␸ ␪ S stretching s( ) are defined from as the positive eigen- value and the direction of the associated eigenvector es . Here, they are given by V. PHYSICAL INTERPRETATION s͑r͒ϭ␧͑nϪ1͒rnϪ2, ͑5.3͒ In the previous sections, the instability was described as ␸ ␪͒ϭϪ ␪ ͑ ͒ a resonance phenomenon of Kelvin modes due to the multi- s͑ n /2, 5.4 polar strain field. We computed the inviscid growth rate of where the angle ␸ is measured in the local polar basis ͓i.e., the resonant combinations and noticed that there were impor- s cos(␸ )ϭe .e ]. For nϭ2, the stretching rate and the direc- tant variations from one resonant configuration to the other. s s r tion of stretching (e ϭe ) are uniform. For larger n, there are For large wave numbers, we showed that only very specific s x both a radial dependence of the stretching rate and an angular resonant configurations were significantly amplified. In this dependence of the stretching direction. section, a physical interpretation is provided, which permits The projected vorticity ␻Ќ of the Kelvin mode one to understand these variations and the inviscid selection v (k,m,␻) is obtained from formulas ͑A10a͒–͑A10d͒.It of the most unstable resonant Kelvin mode configuration. K takes the form For this purpose, it is informative to first analyze the ␻ ϭ ͒ ␺ ϩ ͒ ␺ ͑ ͒ different terms in the linearized inner vorticity equation for Ќ f ͑r sin er g͑r cos e␪ , 5.5 the perturbations: where uץ D␻ ϭ2 ϩe ϫ␻ϩS␻, ͑5.1͒ ␺ϭ ϩ ␪Ϫ␻ ͑ ͒ z z kz m t, 5.6ץ Dt where ␻ is the perturbation vorticity, u its velocity, D/Dt is and f and g are real functions of r. The direction of this the convective derivative and S is the strain tensor of the vector is then given ͑in the local polar basis͒ by a formula of inner basic flow given by ͑in cylindrical coordinates͒ the form cos͑n␪͒ Ϫsin͑n␪͒ 0 g͑r͒ 1 ␸␻ ͑ ␪ ͒ϭ ͩ ͪ ͑ ͒ Ќ r, ,z,t arctan . 5.7 Ϫ ͒ ␺ Sϭ␧͑nϪ1͒rn 2ͩ Ϫsin͑n␪͒ Ϫcos͑n␪͒ 0 ͪ . ͑5.2͒ f ͑r tan 000 Similarly, for a combination of two Kelvin modes ␻ ϩ ␻ The first term on the right-hand side of Eq. ͑5.1͒ represents vK1(k1 ,m1 , 1) vK2(k2 ,m2 , 2) we get ␻ the tilting and the stretching of the basic flow vorticity 0 g ͑r͒cos ␺ ϩg ͑r͒cos ␺ ϭ ␸ ϭ ͩ 1 1 2 2 ͪ ͑ ͒ 2e by the perturbation. The second and third terms are, ␻ arctan , 5.8 z Ќ f ͑r͒sin ␺ ϩ f ͑r͒sin ␺ respectively, the tilting and the stretching of the perturbation 1 1 2 2 vorticity by the basic flow. The role of each term in the with instability has been discussed in various places ͑see, for in- ␺ ϭ ϩ ␪Ϫ␻ ͑ ͒ stance, Refs. 22,24,38͒. Orszag and Patera24 showed that the 1 k1z m1 1t, 5.9 first term alone, or the second term and the third term taken ␺ ϭk zϩm ␪Ϫ␻ t. ͑5.10͒ together, cannot lead to exponential instability. In contrast, 2 2 2 2 the third term alone is sufficient for instability as it provides It is important to note the dependence of ␺ on kz, m␪ or exponential growth of the vorticity component aligned with ␻t in Eq. ͑5.6͒. For fixed r and ␪, this dependence implies 22 the direction of stretching ͑principal axis of the strain ten- that ␸␻Ќ, for a single Kelvin mode, takes all possible values sor S with the largest positive strain rate͒. Here, in addition, as t or z varies and has a mean value independent of ␪, ␸ ␪ one can show that the first and second terms taken together whereas s is periodic. The result is that there is no mean ␸ do not provide instability. Indeed, these terms are associated correlation between the direction of stretching s and the with a solid body rotation ͓␧ϭ0 in Eq. ͑1.1a͔͒: The normal direction of the projected vorticity ␸␻Ќ for a single Kelvin mode solutions are, in that case, nothing more than the neu- mode. This is reassuring as a single Kelvin mode was not tral Kelvin modes studied above. expected to be unstable in a planar strain field.

FIG. 12. Distribution of the angle ␣ between the direction of the vorticity in the (x,y)-plane and the direction of stretching for the principal modes (Ϫ1,2,1) ͑solid line͒,(Ϫ1,1,2) ͑dashed line͒, (1,5,1) ͑dotted line͒ and a ϭ ϭ mode with m1 10, m2 13 but taking the resonant crossing point of the FIG. 11. Example of good correlation between stretching and vorticity di- first and third branches, respectively (᭺) ͑the boxed in mode of Table II͒. rection for an unstable mode of the Rankine vortex in a tripolar strain field (nϭ3). Local direction and intensity of stretching ͑a͒ and projected vorticity in the (x,y)-plane of the principal mode (Ϫ1,2,1) in zϭ0 ͑b͒, z ϭ␲/2k ͑c͒ and zϭ␲/k ͑d͒. the norm of the projected vorticity ͉͉␻Ќ͉͉ as the weight of each point. These curves exhibit a peak near ␣ϭ0 for the unstable modes (Ϫ1,1,2), (Ϫ1,2,1) and (1,5,1) indicating a However, for a two Kelvin mode combination, the above strong correlation between the stretching direction and the argument does not always apply because cancellation in Eq. vorticity. For a ‘‘nonprincipal’’ mode, for instance the one ͑5.8͒ is now possible. In particular, ␸␻Ќ may have a nonzero associated with the first and third branches of the dispersion ␪ ϭ ϭ mean value dependent on at particular values of r if the relation for m1 10 and m2 13, respectively, this peak is ϭ ␣ϭϮ␲ two modes have same axial wave number (k1 k2) and the smaller and there is a secondary peak at /2. The cor- ␻ ϭ␻ ␸␻ ϭ␸ same frequency ( 1 2). If this occurs and if Ќ s for relation is then weaker and the inviscid growth rate much ␴ ϭ ␧ ͒ a specific location at any time, the planar component of vor- smaller ( i 0.0681 from Table I , as expected. ticity is locally amplified by stretching at that position with a These graphs and the above discussion demonstrate that growth rate equal to s(r). Accordingly, the maximum the mechanism of instability is directly related to the local growth rate of the perturbation is always bound by the maxi- stretching of perturbation vorticity by the basic flow. It con- mum local stretching rate of the strain field ͓here (nϪ1)␧͔. firms that the most unstable modes are those which maxi- This upper bound corresponds to a pointwise maximum mize the alignment of their projected vorticity with the local growth rate. It should not be mixed with the local maximum direction of stretching. Moreover, for both nϭ3 and 4, we growth rate obtained in Ref. 29 which is a mean growth rate have seen above that the most unstable combinations tend to along a closed streamline: In general, the local maximum be localized near the vortex core edge where the stretching growth rate is smaller because the perturbation vorticity is rate in the core is maximum. not aligned with the stretching direction on the whole Why principal modes are the configurations which maxi- streamline. Case nϭ4 is an exception: both local and point- mize the alignment of vorticity and stretching is another is- wise maximum growth rates are equal in that case. More- sue. First, note that these configurations have a frequency ␻ ϩ over, we saw in Sec. IV that this maximum is also reached approximatively equal to (m1 m2)/2 so that the ‘‘radial ␦ ␦ by the growth rate of the most unstable resonant Kelvin wave numbers’’ 1 and 2 of the resonant Kelvin modes ͓ ͑ ͔͒ mode combination in the large m1 and k limit. This can be given by formula Eq. A11 are close to each other. This is explained by the fact that, in this limit, the projected vorticity clearly visible in the large wave number analysis of Appen- of the perturbation is localized in the region of maximum dix B, where the most unstable modes have been found to ͑ ϭ ␦ ϭ␦ stretching rate close to r 1) and is everywhere aligned satisfy 1 2 at leading order. Therefore, principal modes with the stretching direction. tend to be more coherent radially than the other modes which For the other unstable Kelvin mode combinations, the could explain why they are the most unstable. A more com- correlation is not perfect but exists as illustrated in Fig. 11. plete explanation has been recently given by Le Dize`s.39 He ␦ Ϸ␦ This figure displays the projected vorticity of the principal showed that the additional condition 1 2 directly results mode (Ϫ1,2,1) ͑see Sec. III for the notation͒, which is un- from the characteristics of the most unstable local modes. In stable in a tripolar strain field (nϭ3). This combination particular, for the elliptical case (nϭ2), he successfully jus- clearly has its vorticity preferentially oriented along the di- tified why the local maximum growth rate is reached when rection of stretching associated with the triangular distortion. this condition is satisfied. Figure 13 illustrates the clear cor- ͉␦ Ϫ␦ ͉ This alignment is quantified in Fig. 12 where the distribution relation of small values of 2 1 with a large inviscid ␣ϭ␸␻ Ϫ␸ of Ќ s is plotted for different resonant modes with growth rate for particular combinations of modes.

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well-described by a weak-strain analysis. However, all studies have restricted the analysis to the stability of these m ϭϮ1 modes12,13,32,40 arguing that other modes should be less unstable. In the present paper, we have shown that this restriction is not justified for the Rankine vortex as the most unstable mode for high viscosity is the principal mode (0,2,1) corresponding to filament bulging–splitting. In a recent paper, Billant et al.37 demonstrated that this bulging– splitting mode was indeed present for a Lamb–Chaplygin vortex pair. For Stuart vortices, they also suggested that the higher order mode, which is briefly discussed by Pierrehum- bert and Widnall,41 and the bulging mode ͑or the so-called Core Dynamics Instability mode͒ examined in Ref. 42 could be related to the principal mode (0,2,1). However, in these ͉␦ Ϫ␦ ͉ FIG. 13. Inviscid growth rate as a function of the difference 2 1 be- three studies, the growth rate of this mode is approximatively ϭ tween the radial wave numbers of the two Kelvin modes m1 10 and m2 half as small as the growth rate of (Ϫ1,1,i) modes ͑bending ϭ 13 for the first 100 crossing points. or translative modes͒. In the present paper, they are almost equal. We have no explanation for this discrepancy except that, in each of these numerical cases, vortices are strongly VI. DISCUSSION deformed and the vorticity is strongly nonhomogeneous. In this paper, the linear stability of the Rankine vortex in To our knowledge, for n-fold symmetry of higher order a weak multipolar strain field was analyzed. This basic flow (nу3), no result concerning the stability is available. This is was shown to be subject to a Kelvin mode resonance insta- due to the fact that the condition of resonance cannot be bility for dipolar (nϭ2), tripolar (nϭ3) and quadrupolar simplified by symmetry as was done by Moore and (nϭ4) strain fields. The unstable modes are combinations of Saffman.13 In principle, for nϭ2, 3 or 4, the condition of two Kelvin modes of the same axial wave number, the same resonance must be analyzed on a case-by-case basis for all frequency, and azimuthal wave numbers separated by n.A modes different from the symmetric principal modes physical interpretation of the instability in terms of local vor- (Ϫ1,1,i) and (Ϫ2,2,i). Note, however, that once resonance ticity stretching was also provided. We demonstrated that the occurs between two neutral Kelvin modes, the above analy- most unstable modes correspond to the modes for which the sis can be formally applied almost without modification. In alignment between the projected vorticity and the direction particular, we expect the main conclusions to hold, i.e.: of stretching is maximized. Viscous effects were also discussed in detail. The critical viscosity above which all ͑1͒ Instability for nϭ2 and nϭ3 for a viscosity smaller than Kelvin mode combinations are damped was computed in an O(␧) critical value and for nϭ4 for viscosity smaller each case. Below this critical value, viscous effects were than an O(␧2) critical value; shown to select a preferential instability mode. For nϭ2, we ͑2͒ stability for nу5 and small ␧; proved that the selected mode is a combination of stationary ͑3͒ the most unstable Kelvin mode combination changes as ϭϪ ϭ helical modes (m1 1, m2 1) if viscosity is sufficiently viscosity varies; and small. However, for larger viscosity, a combination of modes ͑4͒ for vanishing viscosity, the most unstable Kelvin mode ϭ ϭ m1 0 and m2 2 become the most unstable. This unstable combination corresponds to the configuration for which Kelvin mode combination has never been studied before. It vortex stretching is maximized. is associated with the bulging–splitting of the vortex. For n ϭ 3, the selected mode is a time-periodic Kelvin mode com- Considering the similarities between the experimental bination with both frequency, axial and azimuthal wave and numerical observations of vortex filament ϭ numbers increasing as viscosity decreases. For n 4, the destabilization1,2,5 and the form of a vortex subject to a Kelvin mode resonance occurs only for large axial wave Kelvin mode resonance, it is natural to discuss the implica- numbers, so viscosity must be smaller for instability in that tions of the present results in the context of turbulent flows. case. However, as for nϭ3, the selected mode was shown to In turbulence, vortex filaments are strained by the back- have a more intricate azimuthal structure as the viscosity decreases. ground turbulent flow or surrounding vortices. As there is no For vortices with continuous vorticity profiles, instability particular symmetry, they are in general elliptically de- by Kelvin mode resonance is also known to exist. In particu- formed in their core. From the present analysis, we argue that lar, Moore and Saffman showed that the stationary resonance if viscosity is small enough compared to the nonaxisymme- ϭϪ ϭ try of the filament (␯Ͻ0.111␧), the vortex can be subject to of helical modes m1 1 and m1 1 is not dependent on a particular profile and generically leads to the instability of an a Kelvin mode resonance instability. We showed that the elliptically perturbed vortex. Robinson and Saffman40 also most unstable mode was, in such a case, either the combina- showed numerically that for finite strain ͓␧ϭO(1)͔, the sta- tion of two helical modes (Ϫ1,1,2), which produces a planar bility of these particular symmetric modes is qualitatively undulation of the vortex, or the combination of a bulging and

ץ splitting mode (0,2,1) leading to the formation of strands on 1 2 the filament. This behavior was observed in experiments1 D Ϫ Ϫ 00 ␪ץ 2 2 2 and in numerical simulations.5 r r 1 ץ In contrast, this instability does not explain the bursting 2 1 Lϭ⌬ϭ D Ϫ 00, ͑A5͒ ␪ 2 2ץ of vortex filaments which is sometimes observed. Pradeep 2 et al.43 gave an explanation which could be related to a sec- ͩ r r ͪ ondary instability of the vortex deformed by Kelvin modes. 00D2 0 44 Lifschitz et al. first showed that this secondary instability 0000 indeed exists for elliptical flows. Recently, Kerswell45 and Mason and Kerswell46 studied this instability analytically and and numerically in a configuration similar to that in Malkus’ 2 2 2 ץ ץ 1 ץ ץ 1 14 experiment. In this elliptic cylinder geometry, it appears D ϭ ϩ ϩ ϩ . ͑A6͒ 2 2 2 2 ץ 2 zץ ␪ץ r rץ that the principal modes (Ϫ1,1,2), (Ϫ1,1,3) and (0,2,1) are r r unstable and that the growth rate of the secondary instability The relation is of the same order as the primary instability. Secondary instability analysis has never been carried out for a vortex in NϭAϩBϩC, ͑A7͒ an open flow configuration but one would expect that the is also used in Appendix B, where results of Kerswell45,46 to remain qualitatively unchanged. Aϭ 1 nϪ2J ͑ ͒ 2 lr , A8a ץ 1 ACKNOWLEDGMENTS BϭϪ rnϪ1 J, ͑A8b͒ rץ 2 We would like to thank T. Leweke for helpful comments Ϫ͑ Ϫ ͒ nϪ2 Ϫ ͑ Ϫ ͒ nϪ2 on this paper and F. W. J. Olver and M. Kruskal for Eqs. n 1 r i n 2 r 00 Ϫ Ϫ ͑B18a͒ and ͑B18b͒. 1 Ϫinrn 2 ͑nϪ1͒rn 2 00 Cϭ . 2 ͩ ͪ 0000 0000 APPENDIX A: NOTATIONS ͑A8c͒ The operators appearing in Eq. ͑2.1͒ are defined by The velocity–pressure field of the Kelvin mode in the vortex core is defined by 1000 ␻͒ϭ ͒ i(kzϩm␪Ϫ␻t)ϩ ͑ ͒ vK͑k,m, U͑r e c.c., A9 0100 ϭ Jϭͩ ͪ , ͑A1͒ where U (Ur ;U␪ ;Uz ;P) is given by 0010 2m 0000 ͑ ͒ϭϪ ͫ͑ Ϫ␻͒␦ Ј ͑␦ ͒ϩ ͑␦ ͒ͬ ͑ ͒ U r i m J r J͉ ͉ r , A10a r ͉m͉ r m Ϫ␻͒ ͑ ץ ץ Ϫ20 m m U␪͑r͒ϭ2␦J͉Ј ͉͑␦r͒ϩ J͉ ͉͑␦r͒, ͑A10b͒ ץ ␪ץ r m r m ץ 1 ץ 2 0 k 2 ␪ U ͑r͒ϭϪ ͓4Ϫ͑mϪ␻͒ ͔J͉ ͉͑␦r͒, ͑A10c͒ץ ␪ rץ Mϭ ͑ ͒ z mϪ␻ m A2 , ץ ץ 00 2 P͑r͒ϭ͓4Ϫ͑mϪ␻͒ ͔J͉ ͉͑␦r͒, ͑A10d͒ ͪ ץ ␪ץ ͩ z m with J␮ as the Bessel function of the first kind and J␮Ј its ץ ץ 1 1 ץ ϩ 0 ␦ z derivative. The scalar in these expressions is the ‘‘radialץ ␪ץ r r rץ wave number’’ and is defined as Ϫ͑ Ϫ ͒ nϪ2 Ϫ ͑ Ϫ ͒ nϪ2 D1 n 1 r i n 2 r 00 k2͑2ϩmϪ␻͒͑2Ϫmϩ␻͒ 2 Ϫ nϪ2 ϩ͑ Ϫ ͒ nϪ2 ␦ ϭ . ͑A11͒ 1 inr D1 n 1 r 00 ͑ Ϫ␻͒2 Nϭ , m 2 ͩ ͪ 00D1 0 ␳ ␾ Amplitudes and (r) of the displacement aK and potential 0000⌽ K of the Kelvin modes are ͑A3͒ i ␳ϭ U ͑1͒, ͑A12a͒ with ␻Ϫm r

2 ͒␦͉͑ 4Ϫ͑mϪ␻͒ J͉ ץ ץ ϭϪ nϪ1 Ϫ nϪ2 ͑ ͒ ␾͑ ͒ϭ m ͑ ͒ ͑ ͒ ␪ , A4 r i Ϫ␻ ͒ K͉m͉ kr . A12bץ ir ץ D1 r r m K͉m͉͑k

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The dispersion relation D(k,m,␻)ϭ0 is given by ␦ ͱ16Ϫn2 ϳ . ͑B4b͒ ͑ ␻͒ϭ͑ Ϫ␻͒␦ Ј ͑␦͒ϩ ͑␦͒ D k,m, m J͉m͉ 2mJ͉m͉ k n 2 ͑ ͒ ͑ ͒ 4Ϫ͑mϪ␻͒ K͉Ј ͉͑k͒ This expression, in addition to Eqs. B1a – B1c , allows one ϩ m ͑␦͒ ͑ ͒ ͑ ͒ Ϫ␻ ͑ ͒ kJ͉m͉ , A13 to estimate the scalar products appearing in Eq. 3.1 . Upon m K͉m͉ k N ϭA ϩB ϩC writing 2͉1 2͉1 2͉1 2͉1 with the definition Eqs. where K␮ is a modified Bessel function and K␮Ј its deriva- ͑A8a͒–͑A8c͒ for A, B and C, we get, after a careful estima- tive. tion of each term, 8Ϫn2 APPENDIX B: INVISCID ASYMPTOTICS ANALYSIS I ϳI ϳ ␦ s ͑␦ ͒s ͑␦ ͒, ͑B5a͒ 1 2 4␲ 1 m1 1 m2 2 In this appendix the inviscid growth rate of the resonant ϳ Ӷ␦ ͑ ͒ Kelvin mode combination is obtained by an asymptotic J1 J2 1 , B5b analysis in the limit of large axial wave number k and in the 8 J ϳJ ϳ ␦ ͑ ͒ limit of large azimuthal wave number m. Our goal is first to 1͉1 2͉2 ␲ 1 , B5c calculate the limit values observed in Figs. 6 and 10, and then to check that they are in agreement with the local short A ͉ ϳϪA¯ ͉ Ӷ␦ , ͑B5d͒ wavelength analysis.29 For this purpose, asymptotic expres- 2 1 1 2 1 sions for the Kelvin modes are used to evaluate the various B ϳB¯ ϳ ͑ ͒ 2͉1 1͉2 I1 . B5e coefficients of the expression Eq. ͑3.1͒ of the inviscid growth ␴ The last term C ͉ has two different expressions for n rate i . 2 1 ϭ2 and nϭ3:

1. Large k analysis ␦ 9 1 ␦ Ϫ␦ C ϳ¯C ϳ ͵ ͩ 1 2 ͪ 2͉1 1͉2 ␲ cos ␦ x dx In this part, the azimuthal wave number m of the Kelvin 2 0 1 mode is O(1); the limit m→ϱ will be analyzed in the next 9 section. A simple expression for the Kelvin mode may be ␦ ␦ ϭ␦ 1 if 1 2 obtained by using asymptotic estimates for the Bessel func- ϳͯ2␲ , for nϭ2, ͑B6a͒ tions J␮ , K␮ and their derivatives: ␦ ͒ ␦ Þ␦ o͑ 1 if 1 2

2 ␦ ͑ ͒ϳͱ ͑ ͒ →ϱ ͑ ͒ 49 1 ␦ Ϫ␦ 3␲ Jm x ␲ cm x as x , B1a C ϳ¯C ϳ ͵ ͩ 1 2 ϩ ͪ x 2͉1 1͉2 ␲␦ x cos ␦ x dx 4 1 0 1 2 2 JЈ ͑x͒ϳϪͱ s ͑x͒ as x→ϱ, ͑B1b͒ ͑␦ Ϫ␦ ͒ m ␲ m 49 sin 1 2 x ϳ ␦ , for nϭ3. ͑B6b͒ 4␲ 1 ͑␦ Ϫ␦ ͒2 ␲ 1 2 K ͑x͒ϳͱ eϪx as x→ϱ, ͑B1c͒ ͑ ͒ m 2x The expression Eq. 3.1 then reduces to ͉C ͉ where ␴ ϳ 2͉1 ␧ ͑ ͒ i J . B7 ␲ 1͉1 c ͑x͒ϭcosͩ xϪ͑2mϩ1͒ ͪ , ͑B2a͒ m 4 This gives ␲ 9 ␧ ␦ ϭ␦ s ͑x͒ϭsinͩ xϪ͑2mϩ1͒ ͪ . ͑B2b͒ if 1 2 m 4 ␴ ϳͯ16 for nϭ2, ͑B8a͒ i ͑␧͒ ␦ Þ␦ In particular, this gives, when ␦→ϱ and ␻Þm in the dis- o if 1 2 persion relation ͑A13͒, 49␧ ␴ ϳ ϭ ͑ ͒ ␲ Ϫ 2 i for n 3. B8b ͑16 n ͒k 32͑␦ Ϫ␦ ͒2 ␦ϳl␲ϩ͑2mϩ1͒ Ϫarctan , ͑B3͒ 1 2 2 4 n ␦ ϭ ͑ ͒ ␦ Ϫ␦ ϭ Besides, for n 3, using expression Eq. B3 , 1 2 (l1 Ϫ ␲Ϫ ␲ ␴ where l is a large integer which labels the branches. Note that l2) (3 /2), which implies that the maximum i(max) ϭ ␧ ␲2 Ӎ ␧ ͉␦ Ϫ␦ ͉ϭ ␲ l is directly related to the branch label i introduced in Sec. III 49 /(8 ) 0.62 is attained for 1 2 /2. by a relation of the form: lϪiϭ f (m). As also noticed in that section, one expects the largest growth rate to be obtained for 2. Large m analysis the resonant Kelvin mode of same label i. It is then natural to In the limit of large m, the modiﬁed Bessel function K͉ ͉ ␦ ␦ m assume that the radial wave numbers 1 and 2 of the two satisﬁes ␻ ␻ resonant Kelvin modes vK(k,m1 , ) and vK(k,m2 , ) are of Ј ͑ ͒ ͉ ͉ 2 1/2 the same order. This leads to K͉m͉ k m k ϳϪ ͩ 1ϩ ͪ . ͑B9͒ ␻ϳ ϩ ͒ ͑ ͒ K͉ ͉͑k͒ k 2 ͑m1 m2 /2, B4a m m

Therefore, the dispersion relation Eq. ͑A13͒ reduces to 63ͱ7 A ͉ ϳϪA¯ ͉ ϳϪ ͱZ Z , ͑B14f͒ 2 1 1 2 32 1 2 Ϫ͑␻Ϫ ͒2␦ Ј ͑␦͒ϩͫ ͑␻Ϫ ͒ϩ͉ ͉͑ Ϫ͑␻Ϫ ͒2͒ m J͉ ͉ 2l m m 4 m m 7 B ϳB¯ ϳϪ ͱ ͑ ͒ 2͉1 1͉2 Z1Z2, B14g 1/2 8 k2 ϫͩ 1ϩ ͪ ͬJ͉ ͉͑␦͒ϭ0. ͑B10͒ 2 m 49 m C ͉ ϳ¯C ͉ ϳϪ ͱZ Z , ͑B14h͒ 2 1 1 2 4 1 2 where a. First case ␦ È␦ Èm ϭ 1/3 2/3 Ј2͑ ͒ ͑ ͒ „ 1 2 1… Z1 2 m Ai a1 , B15a If ␦ϳm, the Bessel function satisfies JЈ (␦)ӶJ͉ ͉(␦) ϭ 1/3 2/3 2 ͒ ͑ ͒ ͉m͉ m Z2 2 m AiЈ ͑a2 , B15b and the dispersion relation is simplified. This corresponds to NϭAϩBϩC the branches of the dispersion relation for which ␻ϭm in and . This leads to kϭ0 ͑see Fig. 3͒. The case which involves the particular 49 ␻ϭ Ϫ ␴ ϳ ␧, for nϭ3. ͑B16͒ branch leaving from m 1 is treated below. The ad- i 32 equate asymptotic expression for the Bessel function ͓see formula ͑9.3.23͒ of Abramowitz and Stegun’s book47͔ is A similar calculation yields 9 21/3 ␴ ϳ ␧ ϭ ͑ ͒ ϩ␰ 1/3͒ϳ Ϫ 1/3␰͒ ϩ Ϫ2/3͒ i , for n 2. B17 Jl͑m m Ai͑ 2 ͓1 O͑m ͔ 16 m1/3 Þ When a1 a2, the inviscid growth rate of the resonant 22/3 3 ϩ AiЈ͑Ϫ21/3␰͒ͫ z2ϩO͑mϪ2/3͒ͬ, Kelvin mode is smaller. An upper bound is obtained using ͑ Þ m 10 the following relations valid only for a1 a2): ͑B11͒ ϱ ͵ ͑ ϩ ͒ ͑ ϩ ͒ ϭ ͑ ͒ Ai x a1 Ai x a2 dx 0, B18a as m→ϱ with fixed ␰ and where Ai is the Airy function. The 0 ͑ ͒ condition of resonance Eq. 2.6 of two Kelvin modes for ϱ nϭ2 and nϭ3 implies that at leading order ͵ Љ͑ ϩ ͒ Ј͑ ϩ ͒ Ai x a1 Ai x a2 dx ϩ 0 m1 m2 ␻ϳ , ͑B12a͒ 2 a1 6 ϭͩ ϩ ͪ AiЈ͑a ͒AiЈ͑a ͒. ͑B18b͒ a Ϫa ͑ Ϫ ͒3 1 2 ͱ16Ϫn2 2 1 a2 a1 kϳ m . ͑B12b͒ n 1 Relation Eq. ͑B18a͒ yields ͑ ͒ ͑ ͒ C ϳ¯C Ӷ 2/3 ͑ ͒ Inserting Eqs. B12a , B12b in the dispersion relation Eq. 2͉1 1͉2 m . B19 ͑B10͒ also gives Equations ͑B14a͒–͑B14e͒ are still valid, so that, using iden- 2 tity Eq. ͑B18b͒, a1 n ␦ ϳm Ϫ m1/3ϩ , ͑B13a͒ 1 1 1/3 1 ͱ Ϫ 2ϩ 91 2 2 4 16 n 4n ͑B¯ Ϫ ͒͑B Ϫ ͒Ͻͩ 1/3 2/3 Ј͑ ͒ Ј͑ ͒ͪ ͉ I ͉ I 2 m Ai a Ai a . 1 2 1 2 1 2 24 1 2 a n2 ͑ ͒ ␦ ϳ Ϫ 2 1/3ϩ ͑ ͒ B20 2 m2 m2 , B13b 21/3 4ͱ16Ϫn2Ϫ4n It immediately follows that where a1 , a2 are O(1) zeroes of the Airy function. 91 ␴ Ͻ ␧. ͑B21͒ Here the branches are labeled by the zeroes of the Airy i 192 ϭ function. Let us first consider the case a1 a2. In that case, the scalar products appearing in Eq. ͑3.1͒ reduce to Then, for these modes, the inviscid growth rate is roughly three times smaller than the maximum growth rate. Case J ͉ ϳ8Z , ͑B14a͒ ϭ 1 1 1 a1 a2, which maximizes the growth rate, corresponds to the ϭ ϩ J ϳ ͑ ͒ branch crossing points i j 1 where i and j are the branch 2͉2 8Z2 , B14b label for m1 and m2, respectively. ϳ Ӷ 2/3 ͑ ͒ J1 J2 m , B14c 7 63ͱ7 I ϳϪͩ Ϫ ͪͱZ Z , ͑B14d͒ b. Second case ␦ ÈZm with ZÅ1 1 8 32 1 2 „ 2 2 … This second case corresponds to the crossing points of 7 63ͱ7 ϳϪͩ ϩ ͪͱ ͑ ͒ the particular branch of the dispersion relation for m2 and the I2 Z1Z2, B14e 8 32 regular branches for m1. In particular, it corresponds to

Downloaded 18 Dec 2006 to 147.94.57.17. Redistribution subject to AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp 676 Phys. Fluids, Vol. 13, No. 3, March 2001 C. Eloy and S. Le Dize`s

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