Point vortex motion on the surface of a spheroid is studied. Exact dynamical equations from the corresponding Hamiltonian are constructed by computing the conformal metric which induces a modified stereographic projection. As a concrete example, the motion of point vortices at the same latitude (called the point vortex ring) is investigated as an extension of the sphere case. The role of eccentricity to the stability of the rotating motion is analysed. The influence of a pole vortex is also discussed.
The dynamics of discrete point vortices is a classical subject in fluid physics, which has been investigated by many researchers (Newton 2001). Because of its direct application to the geophysical sciences (e.g. to the dynamics of planetary atmospheres and oceans), the motion of point vortices on the surface of a sphere has been of great concern (Kidambi & Newton 1998). Specific topics include various kinds of vortex equilibria (Lim et al. 2001; Jamaloodeen & Newton 2006; O’Neil 2008; Newton & Sakajo 2009), the total (or partial) collapse of vortices (Kimura 1988; Kidambi & Newton 1999; Sakajo 2008), the dynamics with boundary (Crowdy 2006) and chaotic behaviour (Sakajo & Yagasaki 2008a,b). (For more details, we refer the reader to the monograph Newton (2001).)
At this stage of research, it is of interest to consider the point vortex motion on more general surfaces (Kimura 1999; Hwang & Kim 2009), e.g. a spheroid, so as to understand the effect of eccentricity on the vortex dynamics. In fact, the angular eccentricity œ for planets and satellites is generally fairly small, but in some cases it cannot be ignored (e.g. œ≈0.1 for Saturn) and may have some effect on the dynamics. It is with this application in mind that the present study was undertaken.
We remark that, for general compact surfaces, the point vortex equations of motion are first derived in the study of Hally (1980) by an abstract approach without specifying the terms. In fact, to proceed further and to complete the equations of motion, it is essential to find an explicit conformal form of the metric of the surface, which is generally very hard and known only for special cases such as a plane or surfaces of revolution with special reflectional symmetry (including a sphere but not a spheroid), as exemplified in Hally (1980) and Gutkin & Newton (2004).
To the author’s knowledge, there has been no mathematical study of the dynamics of vortices on the surface of a spheroid, except for a recent paper (Castilho & Machado 2008). More precisely, in Castilho & Machado (2008), the problem is formulated as a first-order perturbation of the Hamiltonian about the spherical state and the approximate equations of motion are derived by constructing the approximate conformal metric to examine the dynamics of vortices for the case where two of the principal axes are of the same length.
In the current study, instead of truncating at the first order in the perturbed expression, we calculate an exact dynamical formulation of vortex motion by first solving for the full conformal transformation from the spheroid onto the plane. The corresponding projection is analogous to the stereographic projection for the spherical case. The method used here can be generalized to other surfaces of revolution. In this conformal setting, we derive the governing Hamiltonian formulation and dynamical equations for point vortices on the spheroid.
As an important example and also with geophysical applications in mind, we consider the dynamics of a point vortex ring on a common latitude of the spheroid. This problem originated from the classical work of Thomson (1883), in which the stability of point vortices on the edge of a regular n-gon is analysed in the plane. This study was subsequently generalized and has been applied to various underlying surfaces such as a cylinder (Souliere & Tokieda 2002; Montaldi et al. 2003) and a sphere (Polvani & Dritschel 1993; Boatto & Cabral 2003; Boatto & Simo 2004; Kurakin 2004; Boatto & Simo 2008). The stability of the point vortex ring on a spheroid will be analysed by comparing it with the case of a sphere. Roughly speaking, it is found that as the spheroid becomes prolate the stability region decreases, whereas as the spheroid becomes oblate it increases. We also observe certain interesting critical phenomena for stability in the presence of an additional pole vortex.
2. Equations of motion
(a) A modified stereographic projection
around the z axis. (The cross section with a plane z=c for c2<R2(1+a) is always a circle.) The shape of E2 is determined by the eccentricity of E and, for clarity, we comment on two different definitions of the eccentricity (http://en.wikipedia.org/wiki/Angular.eccentricity). The first is the mathematical notion of eccentricity given by for a>0 and for a<0, respectively. The second is the angular eccentricity (or sometimes called flattening) frequently used in geophysics and is given by for a>0 and for a<0, respectively. These two different notions are, however, both providing the relative degree of difference of the two major axes of an ellipse. More precisely, as a is positive or negative, the spheroid is prolate or oblate, respectively (figure 1). Many planets have small positive but variously different angular eccentricities or flattenings; for instance, œ≈0.09796,e≈0.43165,a≈−0.18632 for Saturn, while œ≈0.0033528,e≈0.081819,a≈−0.0066944 for the Earth.
Let us introduce the usual parametric representation of E2 as 2.3 for 0<θ<π, 0<ϕ<2π representing the latitude and the longitude, respectively. Accordingly, this coordinate patch induces the following metric on E2 2.4 where we easily recover the metric in spherical coordinates of the sphere of radius R if a=0.
On the other hand, as suggested by Hally (1980), to analyse dynamics of vortices on E2, it is very convenient to find a proper conformal (or isothermal) metric of the following form 2.5 which will simplify the subsequent mathematical arguments. However, as remarked by Castilho & Machado (2008), the conformal factor h=h(x,y) is almost unknown except in the case of a sphere. In this paper, we explicitly calculate the conformal factor for the spheroid by extending the spherical derivation via stereographic projection. Let us first recall the spherical case. For the sphere of radius R,
we suppose r=f(θ), ϕ=ϕ to be a (new) parametrization to make the conventional spherical metric 2.6 conformal. (The existence of such f is guaranteed for surfaces including a sphere and spheroid (Hally 1980).) Then, we rewrite equation (2.6) as 2.7 and it should be 2.8 which produces a solution 2.9 for an arbitrary constant C1. Here C1 is a scaling factor and the case of C1=1 corresponds to a special geometric meaning. More precisely, for C1=1, the map represents a stereographic projection of the sphere onto the plane preserving the angle ϕ. From this, we obtain a conformal metric in the projected plane. We also remark that setting C1=1 implies f(π/2)=1, that is, the equator is mapped onto x2+y2=1 in the plane. In fact, this map is geometrically known to be conformal by preserving the angle.
We generalize the method above for the spheroid and explicitly construct a corresponding conformal metric. Writing equation (2.4) as 2.10 we obtain the differential equation for f(θ), 2.11 for conformality. At z=0 (i.e. the equator x2+y2=R2), the spheroid E2 and the sphere S2 coincide, which then produces a necessary boundary condition (Castilho & Machado 2008), 2.12 Assuming f′/f>0, the solution is 2.13
Although defined above piecewise, f(θ) is easily checked to be continuous and smooth in 0<θ<π for every a>−1, a≠0. First, these formulas are valid basically for a>0 and then for a<0; by substituting by in the above expressions, we can easily check that they are still solutions. Therefore, equation (2.13) is correct for any a>−1,a≠0. Moreover, by expanding in a series of a for small but non-zero a>0, we obtain 2.14 which is valid for 0<θ<π. Hence, by letting for fixed (θ,ϕ), we recover the spherical case equation (2.9) for the zeroth-order term and the first-order approximation (Castilho & Machado 2008) for the first-order term.
Geometrically, the transformation above r=f(θ), ϕ=ϕ for a≠0 can be interpreted as a modified stereographic projection of the spheroid E2 onto the plane. We point out that a direct stereographic projection along straight lines of the spheroid onto the plane produces a non-conformal metric on the plane which is different from the case of spherical stereographic projection. Thus, we need a proper deformation of the spherical projection for conformality. The current projection r=f(θ) is a non-uniform dilation in the r direction keeping the argument ϕ same. We also remark that the present approach can be more generalized for closed compact surfaces with certain axial symmetry.
(b) Equations of motion for point vortices on E2
Next, we derive equations of motion for point vortices on E2. Let us suppose that N point vortices zi of strengths Γi are initially placed at (θi,ϕi) for i=1,…,N on E2. In terms of the conformal factor h=h(θ) in equation (2.14) on the projected plane, we first represent the corresponding Hamiltonian governing the motion of point vortices as in Castilho & Machado (2008). In general, by taking a complex coordinate of projected image of the i-th point vortex in the projected plane (and its complex conjugate), the Hamiltonian is expressed by
We explicitly calculate as and rewrite the Hamiltonian conveniently using h=h(θ) as Let us choose the usual canonical variables by recalling the spherical case, then we recover the Hamilton’s equations For convenience, we next introduce new variables zi by (so pi=Γzi) for i=1,…,N in a cylindrical coordinate system (Castilho & Machado 2008) and define a new function g(z) by In terms of z and g(z), we represent the Hamiltonian as 2.15 It is also useful to derive a differential relation for g(z) for later discussion. Differentiating on θ and using equation (2.11), we have the differential identity for g(z) 2.16 We then calculate the velocity of the i-th point vortex in θ, ϕ coordinates as 2.17 2.18 which completes the dynamical equations of point vortices at (θi,ϕi) for i=1,…,N.
3. Latitudinal ring of point vortices on E2
Let us first consider the planar point vortex motion. The special case when N-point vortices all of the same non-zero strength are initially located on the edges of a regular N-polygon is called the N-ring of point vortices. As time proceeds, the vortices are uniformly rotating around the centre with a constant angular speed forming a relative equilibrium (Lim et al. 2001). Ever since the study of Thomson (1883), the dynamical properties (particularly stability) of such an N-ring of vortices has been an interesting topic for many researchers (Havelock 1931; Khazin 1976; Cabral & Schmidt 1999). In the plane, it is found to be stable for N<7 and unstable for N>7 by standard analytical argument. But the case of N=7 (called the Thomson heptagon) was noticed as a peculiar configuration dividing stability and instability, which eventually turns out to be neutrally stable after several decades (see Boatto & Cabral (2003) for details). The ring of point vortices problem is generalized later and is investigated analogously for various underlying surfaces such as a sphere (Polvani & Dritschel 1993; Boatto & Cabral 2003; Boatto & Simo 2004; Kurakin 2004; Boatto & Simo 2008) or a cylinder (Souliere & Tokieda 2002; Montaldi et al. 2003). In the same spirit, here we study the stability of a vortex ring on the spheroid as an application and also a concrete example of previous mathematical formulation.
We suppose the latitudinal ring of N-vortices has the same strength Γi=Γ≠0 initially placed at 3.1 for i=1,…,N, where 0<θ0<π,−1<z0<1 are certain constants. For different values of a with θ0=π/6, we draw the corresponding configurations of a five-point vortex ring on E2 in figure 1.
By the symmetry of point vortices location, the N-vortices are rotating with constant angular speed around the north pole without deforming the initial configuration, thus constituting a relative equilibrium. In fact, we explicitly compute the velocities from equations (2.17) and (2.18) as 3.2 for i=1,…,N and confirm the rigid rotation. (It is interesting that the angular velocity ω0 is independent on a and thus identical to that of the spherical case (Boatto & Cabral 2003).)
There have been a few different approaches to analyse the stability of ring configurations on the sphere (Boatto & Cabral 2003; Boatto & Simo 2004; Kurakin 2004), among which we adopt the one in Boatto & Cabral (2003) for a relatively small number of vortices since it is very simple and straightforward. However, for more vortices, for example N=7 or with a pole vortex, this procedure is too heavy to compute and we try an alternative systematic argument using the property of circulant matrices as first suggested in Boatto & Simo (2004).
Here, we briefly sketch the criterion of stability for the reader (details are in Boatto & Cabral (2003) or Boatto & Simo (2004)). Via the change of variables we reduce the degree of freedom by 1 and obtain a reduced Hamiltonian as during the motion. Next, we recall the Dirichlet theorem (Boatto & Cabral 2003), which states that an equilibrium is stable if there exists a positive (or negative) definite integral in the neighbourhood of the equilibrium of the given Hamiltonian. This implies, since the Hamiltonian is itself a first integral, that the equilibrium is stable if the quadratic form (i.e. Hessian) derived from the Hamiltonian is positive (or negative) definite. We can effectively check the definiteness of Hessian by the method of Jacobi considering the signatures of the minors of the quadratic matrix. More precisely, let Q(u,u) be a quadratic form explicitly given by Further, let the determinants of principal minors be Δ1,…,Δn and assume all to be different from zero. Then there exists a basis relative to which Q is represented as where ζ1,ζ2,…,ζn are the coordinates of u in the new basis. Thus, summarizing, we need to inspect the signatures of the minors of the Hessian of the reduced Hamiltonian to determine the stability. We are primarily interested in the difference with the spherical case, and particular attention is given to the effect of eccentricity on the stability of vortex rings. Computations are performed using Maple and results are compared with those of the spherical case in Boatto & Cabral (2003) and Boatto & Simo (2004).
For N=2 to N=6 this procedure is relatively simple and easily computable. However, for larger N>6, it is almost impossible to directly compute all the principal minor determinants of Hessian matrices since they are too large. To overcome this difficulty, a systematic and comprehensive analysis of the Hessian matrix adopting its special symmetric structure is presented following Boatto & Simo (2004). Briefly speaking, the Hessian is subdivided into four block square submatrices that inherit a particular structure called symmetric circulant. This then enables us to easily diagonalize the original Hessian matrix with explicitly computed eigenvalues and eigenvectors. In fact, we realize that the Hessian matrices of the reduced Hamiltonian are of very similar form to those of a reduced spherical Hamiltonian and follow the procedures of Boatto & Simo (2004, 2008) with a little minor modification for the case of N=7 in this paper.
We will investigate the cases for N=2,…,7 in the following. Roughly speaking, the results show that (i) as the number of point vortices increases, the stability region decreases monotonically and that (ii) as (thus E becomes extremely prolate), the stability region contracts to an empty set for each N. For brevity, we put Γ≡1 always in the computations below.
For two-point vortices we easily compute The stability condition is then p2−a+ap2>0 for p≠1,−1 which is shown as a shaded region in figure 2. We observe that, for negative a (the spheroid is oblate), the vortex motion is always nonlinearly stable since −1<p<1 always, while for positive a (the spheroid is prolate) is stable only for or equivalently , where e is the eccentricity of the ellipse E in equation (2.2). As the stability region contracts indefinitely and the vortex motion eventually becomes unstable except at the poles.
Similarly, we compute the case of N=3 as Since Δ4≥0 always, it should be Δ3<0 for stability if p≠1,−1. So the region for stability is identical to the case of N=2, as in figure 2. On the sphere, this phenomenon is already reviewed in Boatto & Cabral (2003) and, interestingly, we find this is, in fact, true for every a>−1.
We proceed to compute the four vortices cases (see details in appendix A) and obtain the stability region, which is smaller than those of the case for N=2,3. The marginal curve is explicitly In particular, if a=0 we recover the result of Boatto & Cabral (2003). It is also interesting that, for a<−1/4, the motion is always stable at every latitude −1<p<1.
The cases of N=5,6 needs a little more complicated computation (see appendix A), and we find the neutral curve for N=5 and . We found no coinciding or intersecting stability regions for these cases. In fact, regions of stability strictly decrease as N increases for N=4,5,6, as in figure 3.
Since the computation of minors becomes too heavy to carry out, we choose an alternative way to analyse this case. Following the approach of Boatto & Simo (2004), we find that the Hessian is almost in the same structure as that of the case of a sphere. Let us explain this briefly. First, it is more efficient to consider the ring of point vortices in a co-rotating frame with the same angular velocity ω0 since it reduces to a fixed equilibrium. Practically, we perform the rotating coordinate transform and define the corresponding reduced Hamiltonian by then satisfies the Hamilton equations with new conjugate variables .
In these new coordinates, the Hessian evaluated at the equilibrium is of the form which essentially inherits almost the same structure as the Hessian of a spherical point vortex ring (Boatto & Simo 2004). More precisely, Q is the same constant matrix with the spherical case for N=7, and P also, after pulling out the diagonal entries, is decomposed into the sum of a diagonal matrix and a symmetric circulant matrix A where the first row of A is for j=2,…,8. The constant r is given by r4=(1−p2)2 and b/r4 is the diagonal entry of P, which will be explicitly calculated below. To compute the eigenvalues of P, we calculate the two terms, As shown in Boatto & Simo (2008), the stability condition is then where λPmax,λAmin denote the largest and smallest eigenvalues of P and A, respectively. Thus, if a=0 (Thomson heptagon), we observe no stability region for p except p=±1, indicating further intricate analysis (e.g. Cabral & Schmidt 1999). But, in the case of a spheroid, we observe the stability region −1<p<1, i.e. the whole region except the north and south poles for every a<−1/2. It might be more interesting to examine what happens to the stability of vortex motion in the interval −1/2<a, but we do not pursue it further here. At least we confirm that the current analysis produces a correct answer for a<−1/2.
4. The effect of pole vortices
Next, we proceed to analyse the influence of pole vortices. First, we consider one pole (north) case in which a pole vortex with non-zero strength is imposed at the north pole under the ring of point vortex configuration. From the symmetry, the pole vortex then does not move but, as observed in previous works (Boatto & Cabral 2003; Boatto & Simo 2004), it may stabilize or destabilize the rotating motion of ring point vortices. Instead of a comprehensive exposition, we shortly investigate the case of N=3, which will clarify the role of pole vortices. We will briefly comment on general cases too.
(a) Single pole vortex case
Let us suppose that the pole vortex is at the north pole (θ=0) with the strength ΓN+1≠0. We are mainly interested in how the stability regions above change for various values of ΓN+1. The quite systematic and detailed study in Boatto & Simo (2004) is inspiring and we adopt the methodology here for analysis. First, owing to the coordinate singularity of polar representation near the north pole, we introduce Cartesian-like local coordinates (qN+1,pN+1) by defining where (xN+1,yN+1) are the canonical variables for a north polar vortex. Under this setting, we still have the Hamilton equations for i=1,…,N+1, where the Hamiltonian HN is now with H given by equation (2.15) and 4.1 and representing the north pole vortex contribution.
It is more efficient to consider the ring of point vortices configuration in a rotating frame with the corresponding angular velocity, ω, as we did in the N=7 case above, as it reduces the motion to a fixed equilibrium. Practically, we perform the rotating coordinate transform for a constant ω to be determined and define the corresponding Hamiltonian by where then satisfies the Hamilton’s equations with new conjugate variables .
Next, we consider the stability of rotating point vortices by determining the signatures of the eigenvalues of the Hessian of (Boatto & Simo 2004).
(b) N=3 with a polar vortex
The basic situation we discuss here is three vortices with a north polar vortex. For more than three vortices, we observe similar phenomena. To make the discussion precise, we adopt the systematic approach of Boatto & Simo (2008), where the Hessian of the reduced Hamiltonian is explicitly computed and analysed. Direct computation produces the Hessian at the (now fixed) critical equilibrium as where 4.2 and 4.3
Thus, this matrix shares a very similar structure with the Hessian of a spherical case, as in Boatto & Simo (2004). In particular, let us consider the diagonal block submatrices Q,P defined as After pulling out the common diagonal entries, we may rewrite them as with and confirm that A is the same symmetric circulant matrix for N=3 as in Boatto & Simo (2004). Exploiting this symmetry and proceeding in the same way as in Boatto & Simo (2008) (details in Boatto & Simo 2004), we explicitly obtain the analogue stability criterion. More precisely, the eigenvalues of P determine the stability of the rotational motion of vortices whose condition is
(There are some typos in Boatto & Simo (2008). For example, the formula (2.13) on p. 2053 is not correct. Also, the condition (2.14) on the same page is true only after substituting −δ by +δ, and so on.) This explicitly produces for stability. Putting a=0, we recover the result of Boatto & Simo (2008).
A sample case of a=0.5 is shown in figure 4 where the stable region is shaded. As ΓN varies, the stable intervals of p are changing and there seem to occur two critical changes at ΓN=−0.5 and near ΓN=1.549655712. More precisely, for each ΓN<−0.5 there is no stability region near the north pole and the vortex motion is stable only for certain a small neighbourhood near the south pole. On the other hand, for ΓN>−0.5, as ΓN increases, a region of stability near the north pole appears and increases with ΓN and merges with the region near the south pole after ΓN=1.549655712.
For general argument, we investigate by first varying ΓN for a fixed a, and then change a (or the eccentricity e), both of which will give different regions for stability. Roughly speaking, as a decreases, the stability region increases, whereas for increasing a the domain contracts, as in figure 5. This change becomes more distinct as we move away from the north pole. Let us explain this in more detail.
(i) Case of a<−0.5
First, by solving the relation , we have 4.4 for the equation of stability curves. By checking the case a=−0.5 is found to be critical in the sense that for a<−0.5 there never appears a separate stability region near the north pole and the stability region is always connected for any ΓN, which is different from the a>−0.5 case, where the stability region is composed of two disconnected intervals for a certain range of ΓN>−0.5. In another words, the stability region monotonically increases with ΓN from the south pole and becomes the whole spheroid eventually on the surface of the strongly oblate (a<−0.5) spheroid.
(ii) Case of −0.5<a<1
For further values of a, more interesting phenomena occur, which we investigate next. After rationalizing equation (4.4), the denominator is factored into For |a|>1, we inspect the limits of ΓN as to obtain while where sign denotes the signature of −a−1+a2. Thus, we conclude that the stability region drastically changes at the critical values of a=1 and . (For example, the case of a=1.5 is shown in figure 6, where we observe a new branch of marginal curve near p=−1, which is generated from a=1. This will be discussed below.)
More precisely, the stability region composed of two disconnected intervals near the north and south poles appears from a=−0.5 and it increases until a=1 (figure 5). Thus, for −0.5<a<1, the stability region increases from the south pole for large negative ΓN until ΓN=−0.5. But then there appear two stability regions near the north and the south poles and they increase for certain ranges of ΓN>−0.5. On the other hand, if we fix such ΓN, the two stability regions (near the two poles) decrease with a. Eventually, for large ΓN the two regions merge together and the motion becomes stable in the whole spheroid.
(iii) Case of a>1
As a increases from 1, a new unstable region appears near the south pole (p=−1) for large positive ΓN (see the curves for a=1.5 in figure 6). Later this region changes its shape radically right after , as in figure 6. Consequently, near the south pole, there exists a stable region for every ΓN for but, to the contrary, no stable region for every ΓN for . It is also interesting that if , for fixed ΓN positively large, the three-vortex motion becomes unstable in a small strip region in the southern hemispheroid, while it attains a stable change near the south pole. After, for , this peculiar region extends up to the south pole and exists for all real ΓN. In short, the stability of the point vortex ring near the south pole is very sensitive to a near the critical value of . This complicated phenomenon of stability for various a is quite interesting and distinctive compared with the dynamics of vortices upon the sphere. But the physical reason is not so clear.
This research was supported by Chung-Ang University research grants in 2009.
Appendix A. Computation of the determinants of the minors
- © 2010 The Royal Society