## Abstract

We examine conditions for the development of an oscillatory instability in two-dimensional vortex arrays. By building on the theory of Krein signatures for Hamiltonian systems, and considering constraints owing to impulse conservation, we show that a resonant instability (developing through coalescence of two eigenvalues) cannot occur for one or two vortices. We illustrate this deduction by examining available linear stability results for one or two vortices. Our work indicates that a resonant instability may, however, occur for three or more vortices. For these more complex flows, we propose a simple model, based on an elliptical vortex representation, to detect the onset of an oscillatory instability. We provide an example in support of our theory by examining three co-rotating vortices, for which we also perform a linear stability analysis. The stability boundary in our model is in good agreement with the full stability calculation. In addition, we show that eigenmodes associated with an overall rotation or an overall displacement of the vortices always have eigenvalues equal to zero and ±i*Ω*, respectively, where *Ω* is the angular velocity of the array. These results, for overall rotation and displacement modes, can also be used to immediately check the accuracy of a detailed stability calculation.

## 1. Introduction

The problem of determining the stability and dynamics of two-dimensional vortex configurations commonly arises across a wide range of disciplines, including geophysical flows (e.g. Turkington *et al.* 2001), electron plasma columns (Kiwamoto *et al.* 2007) and the study of quantum condensates (Keeling & Berloff 2008). In all of these fields, the stability of coherent vortices is fundamental to the underlying dynamics. Unfortunately, performing a linear stability analysis is often a process substantially more labourious than computing the steady vortex flows. This is epitomized by the fact that the stability properties of several two-dimensional vortex flows have been the subject of protracted debates, in spite of their apparent simplicity. Prominent examples include co-rotating vortex pairs (Dritschel 1995 and references therein) as well as Kármán streets of finite-area vortices (Meiron *et al.* 1984 and several references therein). (More recently, similar debates have developed for ellipsoidal vortices in quasi-geostrophic flows; see Dritschel *et al.* 2005).

For this reason, the development of a simple stability approach (circumventing the need for a complete linear analysis) would represent a particularly useful tool. Such an approach could be used in its own right to obtain basic stability information, or it could be employed in conjunction with more complex stability methods to provide a quick check of accuracy.

To discuss this in further detail, we need to point out that the instabilities experienced by a fluid flow may be subdivided into two qualitatively different categories. The first type is known as an *exchange of stability*, and is manifested by a perturbation that (to linear order) grows exponentially without propagating in space. An example of a fluid phenomenon associated with this instability is given by the merger of two co-rotating vortices, which has been the subject of extensive theoretical, experimental and numerical work in recent years (see Meunier *et al.* 2002; Cerretelli & Williamson 2003 and references therein). The second type of instability is of an *oscillatory* type, and develops through a resonance between two previously stable modes (as we discuss in detail further below; see MacKay 1986; Lamb & Roberts 1998). A classic example of resonance in a two-dimensional vortical flow is provided by the development of the Kelvin–Helmholtz instability, for an otherwise stably stratified fluid, as the velocity discontinuity **Δ***U* is increased; two (previously stable) travelling modes interact to give rise to an instability that propagates while growing (Cairns 1979; Craik 1985). Recently, Luzzatto-Fegiz & Williamson (2010*a*,*b*) developed a simple methodology to detect *exchanges* of stability occurring along a family of steady flows, through the construction of ‘imperfect-velocity-impulse’ (IVI) diagrams. This recent work motivated us to develop a similar, simple diagnostic for predicting the occurrence of *oscillatory* instabilities in vortical flows.

In order to introduce the ideas that we propose to develop, let us first note that when investigating stability of vortex configurations, the focus is typically placed on inertial instabilities (e.g. Eloy & Le Dizès 2001). In the absence of solid boundaries or critical layers, it may therefore be possible to neglect viscous effects; one can then compute families of steady flows, without the need to introduce any external forcing. Furthermore, the lack of dissipation leads to a system that is time reversible and Hamiltonian (see the reviews of Salmon 1988 and Morrison 1998). This Hamiltonian, reversible structure is useful in categorizing the possible stability behaviour, as we discuss below.

When considering linear stability to a perturbation of the form e^{σt}, one consequence of time reversibility is that all complex eigenvalues must appear in quadruplets ±(*σ*_{R}±i*σ*_{I}) (Lamb & Roberts 1998). Special cases are given by *σ*=0, which may occur as a single eigenvalue, and by the purely real or purely imaginary cases (*σ*_{1,2}=±*σ*_{R} or *σ*_{1,2}=±*iσ*_{I}), which can appear as pairs. If all of the eigenvalues are purely imaginary, the system is stable (e.g. Holm *et al.* 1985).

For reversible flows, a different approach to investigating stability can be obtained by considering certain conserved quantities, which impose constraints on the possible dynamics. In essence, this method relies on finding a conserved quantity, say *H* (which may be the Hamiltonian), such that a stationary point of *H* (with respect to kinematically admissible perturbations) corresponds to an equilibrium flow. If one can prove that the equilibrium is a minimum or a maximum of *H*, then the system must be stable, since the growth of a perturbation would lead to a change in *H*, which is impossible. This argument forms the basis of several stability approaches, including Kelvin’s variational argument (see Davidson 1998 and references therein) as well as the method of Casimirs (Holm *et al.* 1985).

Let us now suppose that we are interested in the stability properties of a family of equilibrium flows, which are organized according to a parameter *λ*. We further suppose that we have previously obtained detailed stability information for one specific steady solution within the family (say, *λ*=*λ*_{0}), and that we are interested in detecting any changes in stability that may take place as we traverse the solution series. If a stable flow initially corresponds to a maximum or a minimum of *H*, a necessary condition for a change of stability is that, as we move along the family of solutions, the equilibrium must become a saddle of *H*. If a change of stability does occur, it must manifest itself as an *exchange of stability*: the imaginary eigenvalue pair *σ*_{1,2}=±*iσ*_{I} goes through *σ*_{1,2}=0, after which it becomes a purely real pair *σ*_{1,2}=±*σ*_{R}, as illustrated in figure 1*a* (e.g. Lamb & Roberts 1998). For *σ*_{1,2}=0, a bifurcation to a new family of steady solutions may occur. For a family of steady flows, Luzzatto-Fegiz & Williamson (2010*a*) showed that the creation or destruction of a saddle of *H* can be immediately detected as a turning point in impulse in an IVI diagram, without the need for a more involved analysis.

However, for certain flows, the equilibrium at *λ*=*λ*_{0} may already constitute a saddle of *H*, while still being stable; this allows for the development of an additional type of instability, as explained below. At a saddle of *H*, certain eigenmodes will be associated with an increase in *H*, while others will drive a decrease in *H*. In dynamical systems theory, the sign of the change of *H* associated with a given eigenmode is named the *Krein signature* of the eigenmode (Krein 1950; Arnol’d & Avez 1968; MacKay 1986); its significance can be understood as follows.

At a maximum of *H*, all eigenmodes must have negative signature. However, at a saddle, modes with positive and negative signatures coexist. Two otherwise stable modes with opposite signature may cooperate to give rise to an eigenmode that leaves *H* unchanged (and is therefore able to grow), thus enabling instability. This mechanism is sometimes referred to as ‘Krein resonance’ in the dynamical systems literature (MacKay 1986; Morrison 1998).

If we consider the evolution of the resonance in terms of the eigenvalue behaviour in the complex plane, we have that two purely imaginary eigenvalue pairs meet on the imaginary axis, after which they move parallel to the real axis, giving rise to an eigenvalue quadruplet (as illustrated in figure 1*b*). Since two (neutrally) stable oscillatory modes are destroyed, while an oscillatory instability develops, such a change of stability is also referred to as a *Hamiltonian Hopf bifurcation* or as a *reversible Hopf bifurcation*, by analogy to the Hopf bifurcation of dissipative systems (see Lamb & Roberts 1998 and references therein).

The interpretation of resonant behaviour in terms of Krein signatures has found application in several contexts, including parallel, stratified flows (Cairns 1979; Craik 1985) and steep gravity waves (MacKay & Saffman 1986). However, it appears that resonant instabilities of vortex flows have been almost exclusively studied through detailed eigenvalue calculations; a classic example is given by the elliptic instability of a single vortex in a weak strain field (see the review of Kerswell 2002). As a matter of fact, ideas from Krein’s theory of Hamiltonian spectra have been introduced only relatively recently to the study of this flow (Fukumoto 2003). To the best of our knowledge, the concept of eigenmode signatures has not been used before to study the stability of arrays of several vortices.

We now outline the approach taken in this paper. We suppose that, within a given family of steady flows, there exists a solution (at *λ*=*λ*_{0}) that is well approximated by a collection of uniform vortices, whose separation distances are large in comparison to their core sizes. For this solution, any eigenmodes (together with their signatures) may be readily found through asymptotic approaches. We then distinguish between two qualitatively different types of perturbations. The first type involves pure displacements of each of the vortices, which leave each core shape unchanged, as illustrated in figure 2*a*. Modes of the second type are instead associated with a pure deformation of each core, but do not change the centroid location for any vortex in the configuration (as exemplified in figure 2*b*). In §2, we begin by showing that, for well-separated vortices, pure displacement modes may have positive signature, while pure deformation modes always have negative signature. An important implication is that, in such a two-dimensional flow, an oscillatory instability must involve a pure displacement mode. In §3, we argue that conservation of impulse constrains the behaviour of the eigenvalues for several pure displacement modes. In §4, we make use of this conclusion to show that one or two vortices cannot exhibit a resonance in a two-dimensional flow. Our theory indicates that resonances are possible for three or more vortices. In §5, after computing the mode signatures for three co-rotating vortices, we employ the ideas presented here to construct a simple model to approximately predict the onset of oscillatory instability. In addition, we perform a linear stability analysis for three vortices, and compare these results with predictions from our model. Finally, as a point of interest, we present the detailed stability properties for the three co-rotating vortices.

## 2. Finding mode signatures for well-separated vortices

In this section, we first review relevant concepts from variational fluid mechanics, and introduce the appropriate choice of *H* for vortical flows. As first pointed out by Lord Kelvin (then known as Sir William Thomson), a steady vortex flow realizes a stationary point of the energy, for a given impulse (Thomson 1876). This idea was later given analytical support by the well-known theorem of Arnol’d (1966) (see also Benjamin 1976; Davidson 1998). In mathematical terms, a steady vortex flow in two dimensions corresponds to a stationary point of the functional:
2.1
where *E* is the excess kinetic energy, (*P*,*Q*) the linear impulse and *J* the angular impulse, given by (e.g. Saffman 1992):
2.2
and
2.3
where *ω*,*ψ* are the vorticity and streamfunction, related by **∇**^{2}*ψ*=−*ω*. Since *H* involves a combination of conserved quantities, together with the fixed parameters *U*,*V*,*Ω*, we have that *H* is also a conserved quantity. Since the first variation of *H* vanishes about the equilibrium (i.e. *δH*=0), the second variation *δ*^{2}*H* constrains the evolution of the system, as explained in §1.

In order to define the signature of a specific eigenmode, we suppose that the flow may be represented through *M* independent variables, say, *α*_{i} (where *i*=1,…,*M*). First, we can write the second variation of *H* as (e.g. Dritschel 1985):
2.4
such that **H**_{2} is the Hessian of *H*. Next, in order to find the eigenmodes associated with the linear stability problem, note that the Hamiltonian structure implies that the equations of motion can be cast as (Salmon 1988; Morrison 1998):
2.5
where **∇**_{α} is the gradient with respect to *α* and **A** is an *M*×*M* matrix (known as the symplectic matrix). The linearized dynamics are then given by
2.6
where **A****H**_{2} is the Jacobian matrix of the linearized system. Writing the *i*th eigenmode as , where ℜ denotes the real part, we have
2.7
where *σ*^{(i)} and denote the eigenvalues and eigenvectors of **A****H**_{2}.

The problem of determining signatures for the linear eigenmodes can therefore be stated as follows. For a given flow, we can first compute the Hessian **H**_{2} and the symplectic matrix **A**. The eigenvectors of **AH**_{2} then yield the linear modes *δ**α*^{(i)} (*i*=1,…,*M*), for which the signatures are found from the sign of
2.8
In §2*a*, we employ these ideas to obtain an approach for computing the signatures of pure displacement modes.

### (a) Signature of pure displacement modes

In this section, we are interested in disturbances that involve displacements of *N* well-separated vortices; each vortex may be moved in a different direction by the perturbation. We therefore model this problem by studying a configuration of point vortices, for which the conserved quantities become (e.g. Aref 1983):
2.9
and
2.10
where *Γ*_{k} is the circulation of the vortex located at (*x*_{k},*y*_{k}). The Hamiltonian coordinates correspond to the vortex locations, giving
2.11
such that the system has 2*N* coordinates. The matrix **H**_{2} can be written as
2.12
where **E**_{2},**J**_{2}, are the Hessians of *E*, *J*. (Note that differentiating *P* and *Q* twice with respect to any pair of coordinates yields zero.) After defining , the second derivatives of *E* can be written as
2.13and
2.14
while for *J*, we have
2.15
All the other entries in **J**_{2} are zero.

The above relations, together with an expression for *Ω* (which, for a given equilibrium solution, is a known property) can thus be used to obtain **H**_{2}=**E**_{2}−*Ω***J**_{2}.

Finally, to find the symplectic matrix **A**, first note that the equations can be cast in Hamiltonian form as (Aref 1983; Newton 2001):
2.16
such that the symplectic matrix **A** can be verified to be
2.17
where **G** is an *N*×*N* diagonal matrix given by and **O** is the *N*×*N* zero matrix. Examples showing the application of the expressions above are given in §§4 and 5.

### (b) Signature of pure deformation modes

To compute the signatures of the deformation modes, we prefer to adopt a different methodology, which involves generalizing a result of Fukumoto (2003) to an array of *N* well-separated vortices. For an isolated circular vortex of circulation *Γ*, Fukumoto (2003) computed explicitly the energy change associated with a perturbation of amplitude *ϵ* and azimuthal wavenumber *m*. In the present notation, he found
2.18

A perturbation with azimuthal wavenumber *m*=1 corresponds to a displacement mode, while *m*≥2 yields pure deformations, as illustrated earlier in figure 2. In what follows, we show that a generalization of Fukumoto’s formula is possible, for the case of *N* vortices subject to pure deformations.

We begin by estimating *δ*^{2}*E*. By noting that **∇**^{2}*δψ*=−*δω*, and applying Green’s theorem, we have
2.19
where *A*_{i} denotes the region occupied by the *i*th vortex, and *δ*^{2}*ψ* is the second variation in *ψ* (evaluated at a point ** x**), owing to perturbations that have been applied on all vortices. Let us decompose

*ψ*into the individual contributions from each vortex, such that we can write: 2.20 In the above,

*ψ*

_{j}is obtained by considering the influence of the

*j*th vortex on the flow field through Biot–Savart’s integral: 2.21 where

*x*_{j}denotes coordinates with origin at the centroid of the

*j*th vortex. For , noting that

*x*_{j}

^{′}∼

*a*and

*x*_{j}∼

*L*, we have (see also Saffman 1992): 2.22 where

*Γ*

_{j},

*I*_{j}=(

*P*

_{j},

*Q*

_{j},0) are the circulation and impulse of the

*j*th vortex, and

**=(0,0,1). Since the origin is at the centroid,**

*k*

*I*_{j}=

**0**by construction. Furthermore, under a perturbation of order

*ϵ*to the shape of the

*j*th vortex,

*ψ*

_{j}expands as 2.23 where is the streamfunction contribution from the unperturbed vortex, and , and so on. We now restrict our analysis to pure deformation modes, that is,

*m*≥2. By definition, this type of perturbation leaves

*Γ*

_{j}and

*I*_{j}unchanged. Therefore, equation (2.22) yields 2.24 Hence,

*δ*

^{2}

*E*

_{i}becomes, using the expression of Fukumoto (2003), and letting

*Γ*denote the reference scale for the circulation of the vortices, 2.25 This implies that the change in

*ψ*owing to perturbations on distant vortices makes a negligible contribution to the change of energy on a given vortex, such that each of the

*δ*

^{2}

*E*

_{i}is found by simply considering the self-induced change in

*ψ*.

Finally, we need to assess the contribution to *δ*^{2}*H* from the second variation of the angular impulse. (As mentioned earlier, *δ*^{2}(*P*,*Q*)=(0,0), since the centroid locations are unchanged by the perturbation.) Let us first write
2.26
where *J*_{i} is the impulse contribution from the *i*th vortex, which we suppose has centroid at (*X*_{i}, *Y*_{i}). We expand *r*^{2} as
2.27
where , and (*θ*_{i},*r*_{i}) are local polar coordinates about (*X*_{i}, *Y*_{i}).

As before, we consider perturbations of the form , where , and so on. Substituting this into *J*_{i}, we obtain
2.28
The angular velocity *Ω* is of order (as may be shown, for example, by considering an array of point vortices). Therefore, letting ,
2.29
so that *Ω* *δ*^{2}*J* is negligible. Incidentally, if the *i*th vortex is placed at the location of the overall vorticity centroid, the corresponding is even smaller, as may be expected on simple physical grounds.

Therefore, we finally obtain *δ*^{2}*H*^{(m)}, for *N* well-spaced uniform vortices:
2.30
Therefore, all pure deformation modes have negative signature. An important consequence of the analysis that we presented in this section is that, in a two-dimensional vortex configuration, deformation modes may not coalesce with each other to give oscillatory instability. Therefore, a resonance must involve at least one of the displacement modes (which may have positive signatures).

## 3. Eigenvalue constraints for displacement modes

In this section, we employ simple physical considerations to argue that the conservation of impulse prescribes the eigenvalues of certain displacement modes. In essence, we proceed by considering certain possible perturbations of the flow, and compare the evolution of the perturbed and unperturbed vortex configurations to extract information about the corresponding eigenvalues.

Let us begin by considering a perturbation involving an overall displacement of the configuration. As an example, let us examine a rotating configuration, whose centroid *x*_{c} is moved from (0,0) to (0,*δy*). After this displacement takes place, conservation of linear impulse preserves the new centroid, and the configuration will rotate at a rate *Ω* about the new location, as shown schematically in figure 3*a*. Therefore, in a frame moving with the undisturbed configuration, the perturbation has the appearance of a (retrograde) mode, having constant amplitude and period *T*=2*πΩ*^{−1} (as illustrated in figure 3*b*). We can conclude that an overall displacement mode always has eigenvalue ±i*Ω*. (For a translating configuration, an overall displacement perturbation will of course appear stationary in a frame moving with the undisturbed vortices, thus yielding *σ*=0, as expected.)

We next consider a perturbation that involves an overall rotation of the configuration (by an angle *δθ*, say; see the left-hand example shown later in figure 4 for an illustration). This mode leaves the centroid of the configuration unchanged, as shown in the figure. If the unperturbed configuration rotated at a rate *Ω*, the perturbed one will retain the same angular velocity, but will continue leading the motion of the undisturbed configuration by the angle *δθ*. If we observe the flow in a frame of reference rotating with the undisturbed vortices, the perturbed configuration will appear stationary; this implies that the corresponding eigenvalue must be zero. Therefore, an overall rotation mode must be associated with a zero eigenvalue.

Furthermore, we note that, if the undisturbed configuration originally translated, the perturbed one will now move along a straight path at an angle *δθ* from the original one. Therefore, the distance between the perturbed and unperturbed configurations will increase linearly in time. Since this corresponds to an algebraically growing instability, the associated eigenvalue must be zero also in this case.

To summarize, an overall displacement mode must have eigenvalue ±i*Ω*, while an overall rotation mode is associated with a zero eigenvalue. We should stress that this result does not rely on any approximation, and must therefore hold for any steady vortex flow.

Note that several investigators have previously commented on the properties of these eigenvalues, in the context of specific vortex flows. Havelock (1931) obtained the same expressions for these two eigenvalues in the case of *N* equal-strength *point* vortices arranged in an equilateral polygon. Kamm (1987) and Dritschel (1985) also remarked that the overall rotation mode from their linear stability analyses appeared to have eigenvalue equal to zero; furthermore, Kamm (1987) observed that the magnitude of the overall-displacement eigenvalue that he computed numerically always seemed to match the value of the angular velocity of the configuration. However, in spite of the simplicity of the arguments used here, the fact that an overall displacement mode, for a general steady vortex configuration, *must* have eigenvalue ±i*Ω* does not seem to have appeared in the literature before.

Finally, it is important to address here the related (yet somewhat distinct) point that, when discretizing an Euler flow, one may expect the presence of a large number of zero-eigenvalue modes, which follow from the degeneracies inherent in the equations (as argued by Crowdy 2002; Elcrat *et al.* 2005). Specifically, these zero eigenvalues follow from the fact that if a particular vorticity distribution constitutes a steady solution, another flow obtained through a rotation or translation of the configuration also yields an equally valid solution. As a consequence, Crowdy (2002) found zero eigenvalues associated with *both* overall rotation and displacement modes. However, as already noted by Crowdy (2002), these zero eigenvalues should not be regarded as physically relevant to the actual dynamics. As a matter of fact, in our numerical work (which is presented in detail further below), we found that a zero-eigenvalue overall displacement mode appeared *in addition* to the mode with eigenvalue ±i*Ω*.

The constraints on the eigenvalues of pure displacement modes discussed above have two immediate applications. Firstly, one may use these results to quickly check the accuracy of a linear stability analysis. The second application involves the analysis of possible resonance scenarios, and is discussed in §4.

## 4. Stability of two vortices

### (a) Signature of pure displacement modes for two vortices

We now illustrate a significant consequence of the theory described above, namely, that one or two vortices may not undergo a resonant instability in a two-dimensional flow. The signatures of the pure displacement modes can be obtained by considering two point vortices with strengths *Γ*_{1},*Γ*_{2}, separated by a distance 2*L*. It is a classic exercise to show that the resulting flow is always an equilibrium configuration, with angular velocity
4.1
Following the analysis in §2*a*, we can compute the Hessians of *E*,*J* and, therefore, **H**_{2}. This is found to have eigenvalues and eigenvectors:
4.2
Notice that *Γ*_{1}*Γ*_{2}<0 implies *μ*^{(2)}<0, and the equilibrium is not a minimum of the energy any longer; however, we will see below that the configuration is still spectrally stable. We must also point out that *μ*^{(4)}≡0 is always associated with a pure rotation mode. This is to be expected, since the system is invariant under rotation; furthermore, we should note that this invariance requires that a pure rotation mode must yield *δ*^{2}*H*=0, independently of the vorticity distribution considered. (Note, however, that the eigenmodes of **H**_{2} are, in general, different from the eigenmodes of the linear stability problem.)

After computing the symplectic matrix **A**, the eigenvalues and eigenvectors of **A****H**_{2} can be found as
4.3
where *σ*^{(3)}=*σ*^{(4)} is a repeated eigenvalue. Note that *σ*^{(1,2)}=±i*Ω*, as predicted by the argument in §3.

Hence, gives
4.4
Therefore, we find that, for two vortices, all displacement modes have non-negative signatures. The overall displacement mode is the only eigenmode that may have positive signature; hence, an oscillatory instability requires the interaction of the overall displacement mode with a deformation mode. One may expect instability to occur when these two modes have the same eigenvalue. However, since the eigenvalue of the overall displacement mode must be equal to ±i*Ω*, this eigenvalue cannot leave the imaginary axis, and may not produce a resonance. This implies that an oscillatory instability cannot occur for two vortices.

We can test this deduction by briefly reviewing existing stability results for one or two vortices. An isolated circular vortex is stable (in both a linear and nonlinear sense; see Thomson 1880; Dritschel 1988). Love (1893) showed analytically that an elliptical vortex becomes unstable exclusively through exchanges of stability. More recently, Dritschel (1995) studied numerically the stability of unequal-area, co-rotating and counter-rotating vortex pairs, without finding evidence of any resonances. Furthermore, Meunier *et al.* (2002) examined the linear stability of *non-uniform* co-rotating vortex pairs with different vorticity profiles, finding exchanges of stability but no resonances. In summary, to the best of our knowledge, all existing data support the conclusion that one or two vortices will first become unstable through an exchange of stability.

## 5. Stability of three co-rotating vortices

Our theory indicates that an oscillatory instability is in principle possible for three co-rotating vortices. As a simple example, we consider three equal-area vortices, each having circulation *Γ*, arranged at the corners of a triangle.

Before beginning the analysis, we introduce the notation used to denote the different eigenmodes that are obtained from the linear stability problem. A given mode is associated with a deformation of azimuthal wavenumber *m*, which occurs simultaneously on all vortices, with a phase increment *ϕ* from one vortex to the next (moving in a counterclockwise sense). For *N* co-rotating vortices, the phase increment takes the form *ϕ*=2*πn*/*N*, where *n* ranges from 1 to *N*; a given eigenmode exhibits a fixed phase angle *ϕ*. We therefore can denote a mode with wavenumber *m* and phase increment 2*πn*/*N* as *m*_{n}. As an example, pure displacement modes labelled according to this approach (denoted as 1_{1},1_{2} and 1_{3}) are shown in figure 4, for three vortices. Additionally, we may display information on a mode’s signature by means of a superscript. Modes having positive, negative or zero signatures are, therefore, labelled as , respectively, thereby building on notation introduced above (for example, *m*_{n}), or on the equally valid one used by Dritschel (1985) (for example, *m*/*n*).

### (a) Signatures of pure displacement modes for three vortices

For three point vortices, the angular velocity is given by (e.g. Aref 2009)
5.1
where, as before, *L* is the distance between each vortex and the vorticity centroid (such that the centroid is at (0,0) and the first vortex is at (*L*,0)), and the eigenvalues and eigenvectors for the linear stability matrix **AH**_{2} are
As for two vortices, we have the repeated eigenvalue *σ*^{(1)}=*σ*^{(2)}=0. (This is of course consistent with the classical results of Havelock 1931.) Notice that these perturbations correspond, in the notation introduced above, to modes 1_{1}, 1_{2} and 1_{3}, respectively (see figure 4 for an illustration). For three vortices, 1_{1} and 1_{3} are the overall rotation and overall displacement modes, respectively.

By using the same procedure as in §4, we find the signatures from
5.2
which leads us to denote the modes as and . As for two vortices, the overall displacement and overall rotation eigenmodes have positive and zero signatures, respectively. However, now represents an *additional* positive-signature eigenmode, which is free to cooperate with a negative-signature eigenmode to give rise to an oscillatory instability.

### (b) Elliptical model to predict the onset of oscillatory instability

Elliptical models of varying degrees of complexity have seen extensive development and use across several types of vortex flows (see Dritschel & Legras 1991 and references therein). In this section, we employ the elliptical model to provide a simple predictive tool for the onset of oscillatory instability.

We parametrize the family of solutions through the separation distance between the vortices, which is quantified by the parameter *r*_{1}/*r*_{2}, as defined in figure 5*b*. Our goal, therefore, is to predict the approximate value of *r*_{1}/*r*_{2} for which resonance develops, without performing a linear stability analysis or computing the full numerical solutions.

The model can be outlined as follows. Firstly, we construct an approximate solution for this flow by representing each vortex as an ellipse (similarly to the work of Saffman & Szeto (1980) for two vortices). Secondly, we employ the closed-form dispersion relation of Moore & Saffman (1971), for an elliptical vortex subject to strain and rotation, and plot *approximate* eigenvalues as a function of *r*_{1}/*r*_{2}. When the frequency curves for two opposite-signature eigenvalues intersect, we expect a Hamiltonian Hopf bifurcation to take place.

The construction of the elliptical steady solutions, for an array of three vortices, involves a conceptually straightforward extension of the two-vortex solutions of Saffman & Szeto (1980), and is therefore not reported in detail here. The angular velocity *Ω*/*ω* that we obtain from the elliptical model is shown by the dashed line in figure 5*a*, and will be compared with the full numerical solution in the next section.

After computing the ellipse axis ratio *λ*=*b*/*a* and the angular velocity *Ω*/*ω* for each *r*_{1}/*r*_{2}, we obtain approximate eigenvalues as follows. For sufficiently large *r*_{1}/*r*_{2}, the dependence of the eigenvalues on the phase angle *ϕ* can be assumed to be weak. Hence, we expect that the eigenvalues for, say, modes 1_{1},1_{2},1_{3} will be approximately the same. We therefore choose to approximate the eigenvalues using the dispersion relation for a single ellipse subject to strain and rotation, which was found by Moore & Saffman (1971) as
5.3
The eigenvalues obtained from the elliptical model are therefore labelled as , and are plotted in figure 6*a*. The accuracy of the resulting prediction of *r*_{1}/*r*_{2} for the onset of resonance is discussed in the next section.

### (c) Stability of three vortices and comparison with elliptical model

In order to test the prediction from the elliptical model for three vortices, we computed numerically the steady states and performed a linear stability analysis. The numerical method that we used to compute the equilibrium flows employs a novel discretization, together with Newton iteration, to accurately resolve vortices of arbitrary shape with an affordable computational cost. Once a steady solution is found, we evaluate the Jacobian associated with the linear stability problem, thus obtaining the eigenvalues and eigenvectors. Further details on the numerical method may be found in Luzzatto-Fegiz & Williamson (submitted). The vortex equilibria studied here were computed accurately to at least seven significant figures.

We begin by assessing the accuracy with which the elliptical model represents the vortex equilibria. To this end, we plot the angular velocity, as a function of *r*_{1}/*r*_{2}, for both the elliptical model and the full solution (figure 5). Remarkably, the value of *Ω* from the elliptical model appears to be quite accurate down to *r*_{1}/*r*_{2}≈0.1, where the error in *Ω* is still only about 2.3 per cent. For smaller *r*_{1}/*r*_{2}, the accuracy of the elliptical model worsens rapidly.

We now discuss the elliptical model’s ability to estimate the location of the oscillatory instability, as shown in figure 6*a*. The eigenvalue curves for modes 1^{+}_{E} and 2^{−}_{E} intersect at . We find that the actual eigenvalue plot for modes and (shown in figure 6*b*) does indeed exhibit a bubble of instability centred approximately at *r*_{1}/*r*_{2}≈0.19. (The onset of the instability occurs at .) The vortex shapes corresponding to the onset of resonance are shown in the insets in figure 6. The elliptical model is therefore quite effective in predicting the onset of an oscillatory instability. Furthermore, we note that the effort required in constructing the elliptical model is small, especially if contrasted with the labour required to compute accurately the steady states, and to subsequently perform a linear stability analysis.

For completeness, we also report the full stability properties for the three vortices, as obtained from our linear stability analysis. The eigenvalue plot (shown in figure 7) exhibits a very rich structure; we remark here on its main features.

As already noted above, the first instability (as *r*_{1}/*r*_{2} is decreased) is associated with a resonance between modes and . Shortly thereafter (at ), eigenmode undergoes an exchange of stability, turning into . For , is the unstable mode with the largest growth rate.

The theory of eigenmode signatures presented here can be used to interpret the subsequent development of other resonances, as *r*_{1}/*r*_{2} is reduced. For example, after the first instability ‘bubble’, modes and separate, leaving free to interact with , leading to another resonance (as shown in figure 7).

Another remarkable fact from figure 7 is that the eigenvalue for mode touches *σ*=0 at , after which it *returns* to being purely imaginary (instead of becoming real and undergoing an exchange of stability, as one might expect). This is shown in detail in the close-up in figure 8. At slightly lower *r*_{1}/*r*_{2}, this eigenmode interacts with to yield another resonance (which, however, maintains a small growth rate). By the theory of Krein signatures discussed in §1, we can therefore infer that must have changed its signature (becoming ) when touching *σ*=0.

This observation brings us to the point that, for vortices that are relatively close together, further resonances may *also* occur if a deformation acquires positive signature by going through *σ*=0 without an exchange of stability. These resonances are, of course, not captured by the elliptical model discussed above.

Incidentally, when the eigenvalue for mode 2_{3} touches zero, the family of steady vortices studied here connects to a series of lower symmetry flows through a transcritical bifurcation. Although these results are not shown here, we were able to resolve this bifurcation in detail using the IVI diagram approach discussed in Luzzatto-Fegiz & Williamson (2010*a*,submitted).

Finally, it is essential to recognize that the properties of *N* co-rotating uniform vortices (including the *N*=3 case) were first examined in the seminal study of Dritschel (1985). In this classic paper, Dritschel broke new ground by numerically computing both the shapes and the linear stability of the *N*≥3 vortices, revealing a very rich eigenvalue structure. However, we must address the fact that our linear stability results (shown in figure 7) present a number of differences from those computed by Dritschel (1985). For example, Dritschel observed that eigenvalue coalescence could occur only between modes with the same phase angle. However, in our results, we observe eigenvalue coalescence also between modes with different phase angles (but which have the same wavenumber *m*), such as and . In our data, the key factor in determining allowable resonances seems to be the signature of the eigenmodes.

We can tentatively explain this and other differences with the work of Dritschel (1985) as follows. By the arguments presented in §3, the eigenvalue for the overall displacement mode must be ±i*Ω*. We can use this as a consistency test for the stability results. Figure 9*a* shows the eigenvalue of the overall displacement mode (as a solid line), together with selected values of *Ω*, adapted from fig. 6 and table 1 of Dritschel (1985). While *σ*_{I} and *Ω* are close for larger *r*_{1}/*r*_{2}, they progressively diverge as *r*_{1}/*r*_{2} is reduced and the vortices become more elongated, and thus are more expensive to compute accurately. Figure 9*b* shows *σ*_{I} and *Ω* for the calculations presented here; the two quantities are indistinguishable up to six significant figures.

## 6. Conclusions

In this paper, we build on concepts from the study of Hamiltonian dynamical systems, and develop a theory of resonant instability in two-dimensional vortex configurations. We show that, for well-separated vortices, deformation modes always have negative signature, while displacement modes may have positive signature. A resonance requires the interaction of opposite-signature eigenmodes, and therefore must involve at least one displacement mode. By employing simple physical arguments, we show that the eigenmodes associated with an overall rotation and an overall displacement of the vortices must always have eigenvalues equal to zero and ±i*Ω*, respectively. Since, for one or two vortices, these are the only possible displacement modes, these constraints on the eigenvalues imply that a resonance cannot occur. All available stability data in the literature, for one or two vortices, support this theoretical finding.

Our theory indicates that a resonant instability is possible for three or more vortices. For these more complex flows, we propose a simple elliptical model to estimate the onset of resonance. We take, as an example, three co-rotating vortices and compare the results from our model with those from a full eigenvalue calculation, finding good agreement.

We must be careful to point out that one further, *indirect* route to resonance exists, for small separation distances between the vortices. The eigenvalue of a deformation mode may go through zero and remain purely imaginary, while the eigenmode acquires *positive* signature. This perturbation may then interact with the negative-signature deformation mode to give an oscillatory instability, as shown in the close-up of figure 8. The mechanism that allows such an indirect resonance deserves further study. Nevertheless, we can note that, as one progresses along an initially stable family of solutions, the eigenvalue of a negative-signature mode will necessarily cross the path of a positive-signature eigenvalue *before* being able to reach *σ*=0 (this is immediately apparent, for example, by examining the path of the eigenvalues in figure 6*a*, as *r*_{1}/*r*_{2} is reduced). This suggests that a direct resonance, when possible, will occur well before an indirect one, as one brings the vortices close together.

In summary, we find that the Krein signature theory presented here constitutes a valuable tool in studying resonant instabilities in two-dimensional vortex configurations. Further to this, our results can be used in combination with more involved stability approaches, to interpret, and compare with, detailed eigenvalue data.

- Received August 27, 2010.
- Accepted October 14, 2010.

- This journal is © 2011 The Royal Society