1. Introduction

The present paper investigates the self-induced velocity field of a uniform vortex having Schwarz function of its boundary with two simple poles on a suitable transformed plane. This analysis is viewed as a first step towards the study of the dynamics of such a vortex in terms of the related Schwarz function.

The use of the Schwarz function in searching vortex equilibria and also in investigating their stability properties dates back to the eighties of the last century (e.g., see the elegant discussion in [1, Section 9.2]). As well known, the Schwarz function of an analytic curve in the plane of the complex variable satisfies the request (a definition is indicated by joining the two symbols “” and “”, with the first one on the side in which the new quantity appears) in any point and is defined via analytic continuation on a suitable neighbourhood of the curve In general, singularities of the function are found inside and outside An extensive theoretical background on the Schwarz function can be found in [2], while an attempt to summarize the original ideas on the use of such a function in searching vortex equilibria may be the following one.

The curve is taken as the boundary of a uniform vortex (with unitary vorticity, for the sake of simplicity) and it is assumed that can be rewritten as the sum of two functions: , where and are analytic inside and outside , respectively. It is worth noticing that the behavior of for follows just from the above splitting of Indeed, consider the integral on the path of Figure 1 of the function , for fixed away from the vortex. It gives the value of in terms of the moments of the Schwarz function: with the integral having in place of vanishing, because is analytic inside By accounting for that the integral for gives times the area of the vortex, the asymptotic behavior of follows as

Figure 1: Integration path for studying the asymptotic behavior of the function

It can be noticed that the conjugate of the velocity is analytic outside , while is analytic inside the vortex (both cases will be unified in terms of a Cauchy integral of in (2.9) below); it follows that must have the form Indeed, the continuity across the vortex boundary follows by that on the , behaves at infinity as and the corresponding vorticity is uniform and unitary inside and vanishes outside. At this stage, it is also easy to show that the asymptotic behavior of the velocity is related to the (complex) moments of the Schwarz function, through the power series of (1.1).

The formula (1.2) relates velocity and Schwarz function in a very appealing way. However, the identification of the functions and is often a quite complicated goal, which is reached by firstly identifying through its asymptotic behavior (1.1) and then by taking as In the present paper, an approach which is alternative to the use of the relation (1.2) is proposed: and are related through a Cauchy integral, which is evaluated without splitting the Schwarz function. The two approaches are completely equivalent: if the Schwarz function is not too complicated, the splitting (1.2) is still more convenient, but in other cases the present approach can give some advantage. As an important example, for the class of vortices investigated in the present paper the use of the integral approach results in being much easier than the one of the splitting (1.2).

When the shape of the uniform vortex is not far from a circular one (quasicircular vortex), the identification of and is rather simple and the analysis can go ahead [3]. The vortex boundary belongs to a ring inside which the Schwarz function is evaluated via a Laurent series, with and being the analytic continuations of its regular and singular parts, respectively. Due to the small differences from a circle, it is also convenient to use a Laurent series in order to define the vortex shape through a conformal map between the unit circle and The coefficients of the series for are nonlinear combinations of the ones of the map. Configurations of absolute equilibrium in presence of external rotation and stationary strain (intensity and principal axes of which are kept fixed in time) have been investigated by imposing that the normal velocity of the fluid at the vortex boundary vanishes: This relation is enforced to be identically satisfied in , that is, any power of is multiplied times a vanishing coefficient: a nonlinear system in the map coefficients follows, to which conditions concerning the area of the vortex and the position of its center of vorticity are added. The system is numerically solved via a multidimensional Newton solver. Finally, the linear stability of stationary solutions also is investigated with the above approach. The coefficients of the conformal map are taken as functions of time, and it is enforced that the normal components of the velocity of the fluid and of the vortex boundary agree: The above constraint is linearized around a stationary solution, time dependence is fixed in the form (with being a complex number, the real part of which determines the stability properties of the vortex) and an eigenvalue problem is deduced. Several interesting cases of bifurcations of equilibria are discussed, with particular regard to the energy conservation. Also the dynamics of quasicircular vortices has been investigated [4], but by using a quite different approach. The vortex boundary is described in terms of a Fourier series and an equation of evolution for the Fourier coefficients is deduced. It is also found that the derivatives along the vortex boundary of the velocity and of the Schwarz function are related through a Cauchy integral.

In the last ten years, a large research activity has been devoted to find and analyze stationary solutions of the Euler equation by using the Schwarz function. The vortex shape is searched by starting from the following form of the streamfunction in a corotating frame of reference: The form (1.3) is assumed valid outside or inside the vortex, while is taken as vanishing on the complementary part of the physical plane. Multipolar equilibria have been analyzed in [5], where symmetrical configurations of point vortices are placed into a uniform one, in such a way that the total circulation vanishes. In this paper, the form (1.3) is built as an extension to general shapes of the streamfunction for the shielded Rankine vortex (see also [6], in which the discussion is carried out in the framework of the quadrature domains by using a fundamental theorem of Aharonov and Shapiro). The stability properties of the above equilibria have been analyzed in [7], both in the framework of a linear theory and through fully nonlinear numerical simulations with a contour dynamics approach (see (2.2) below). Quadrupoles, pentapoles, and higher have been found to be stable equilibria, while the tripoles result in being linearly unstable. Other interesting equilibria involving a doubly connected uniform vortex and an internal set of pointwise ones are found in [8], mimicing the overlapping of shielded Rankine vortices. An irrotational region remains trapped at the center of the vortex. In [9], a class of stationary solutions consisting in a uniform vortex surrounded by a certain number of point corotating vortices placed on the vertices of a regular polygon is built and analyzed. This vortex pattern depends on an integer number , which specifies the number of satellite vortices, and on a parameter belonging to a lower bounded interval of the real axis. In correspondence to the minimum value of and for a given , called , the central uniform vortex exhibits a nonanalytic boundary in which cusps are present. For growing , the central uniform vortex becomes smaller and smaller (its area decreasing as ), with its shape going to a circular one. In [10], the streamfunction (1.3) is used to generalize to finite-area vortices the ideas of Aref and Vainchtein [11] who search asymmetric equilibria of point vortices by inserting new vortices on points of relative rest (in a corotating reference frame). By such an approach, growing uniform vortices are inserted in a corotating vortex pair until the Rankine vortex is reached. The stability of such one-parameter family of equilibria has been also tested through numerical simulations with a contour dynamics approach. Other equilibria depending on two parameters and involving uniform vortices and point vortices are found in [12], still starting from the streamfunction (1.3). A central uniform vortex is surrounded by an alternate distribution of pointwise and uniform vortices. Also in this case, vortex shapes with cusps are found for certain critical values of the parameters and numerical simulations show the formation of filaments in configurations having large satellite vortices with cusps, as well as in perturbed (by displacing point vortices) equilibria.

The present paper deals with the self-induced velocity of any vortex, the boundary of which possesses a Schwarz function with two simple poles (on a suitable transformed plane). This vortex shape appears to be the simplest, but nontrivial, possible one: the velocity can be evaluated analytically by using an integral link with the corresponding function of its boundary. For this reason, it has been selected by the authors as the starting point of an analysis of the vortex motion through the dynamics of , which satisfies the evolution equation [13]: In order to investigate nontrivial sample cases, the analytical forms of the velocity are needed in (1.4). The present analysis achieves the first step of such a way. At the same time, it shows the advantage of using an integral link between and

The paper is organized as follows. The integral relation between Schwarz function and velocity is presented in Section 2. In Section 3, an overall view of the geometrical properties of the vortices having Schwarz function with two simple poles is given. A classification of this kind of vortices is proposed in Section 4, while a discussion of several sample cases follows in Section 5. In Section 6, the inverse map (from the physical plane to the transformed one) is built and the self-induced velocity is analytically evaluated. The different velocity fields (together with the corresponding streamfunctions) are shown and discussed in Section 7, where we also propose the use of an equivalent Rankine vortex in order to investigate the dependence of the velocity on the vortex shape. Conclusions are offered in Section 8, together with a sketch of the principal research lines under investigation at the present time.

2. An Integral Relation between Schwarz Function and Velocity

The velocity induced by a uniform vortex (inside which the vorticity is unitary, for the sake of simplicity) depends only on the shape of its boundary : if it is smooth and its length is finite, then in any point of the plane the velocity is given by where is the Green function of the Laplace operator, that is, (the modulus of a vector is represented with the same symbol, but without using the bold character, e.g., ), and is the curve element. The motion of such a kind of vortex is defined by the time evolution of its boundary, starting from a smooth and finite-length boundary at the initial time (). In order to numerically investigate that motion, the form (2.1) of the velocity is used as briefly described below.

Consider a parameter on at time With the vortex motion being a material one, it is possible to write as a function of the corresponding parameter at the initial time () and of the time itself: The Lagrangian representation of the position on the vortex boundary becomes natural and the velocity (2.1) evaluated on that point, that is , gives the Lagrangian velocity It follows that the motion of is the solution of the Cauchy problem: The above approach is known as contour dynamics [14, 15], a powerful tool to investigate the inviscid, incompressible two-dimensional vortex motion. Obviously, in the numerical practice a certain number of nodes are selected on the initial boundary and their motion is followed in time. In the discrete framework, the velocity (2.1) is evaluated through an interpolation procedure which rebuilds an approximation of

By starting from the contour dynamics form of the velocity (2.1), in the present section an integral relation between Schwarz function and velocity is deduced, which is equivalent to the form (1.2) but it does not require the splitting of the Schwarz function in the sum The key observation about (2.1) lies in considering that if the point is not on the vortex boundary, an integration by parts enables us to write with the dot indicating a scalar product. By conjugating both sides of the above relation, it may be rewritten in the following complex form: Notice that the first integral in the right-hand side of the above equation holds if lies inside the vortex (), while it vanishes when is external to the vortex (). Equation (2.4) relates velocity and Schwarz function in any point which does not lie on the vortex boundary, while if the point belongs to that curve, the discussion must be carried out in a more sophisticated way.

The tangent derivative of the conjugate of the velocity on the vortex boundary has been investigated in a previous paper [4], where it has been shown that it is related to the tangent derivative of the Schwarz function, that is, to the function , with being the unit vector which is tangent to The relation is through a Cauchy integral (see [4, equation (18)], rewritten here for reader's convenience): By changing with , an integration by parts gives in which the second integral in the last side holds , it follows that In this way, the conjugate of the velocity in the point belonging to the patch boundary becomes

Equations (2.4) and (2.8) are summarized by introducing the characteristic function of the domain , which holds inside , outside , and on the boundary, in the following new form of the self-induced velocity: with the position that the integral must be a Cauchy one if the point lies on the curve It is worth noticing that the velocity (2.9) is a function of both and inside the vortex, while it depends on only on (through the Schwarz function) and outside the vortex. The velocity (2.9) is a continuous function across : indeed if approaches from the inside of the vortex, goes to and the integral to plus continuous terms (given by the singularities of the Schwarz function), while if reaches from the outside, the first term vanishes and only the above continuous contribution remains. On the vortex boundary, holds , while the integral must be considered as Cauchy's one and it leads to a contribution plus the continuous contribution due to the singularities of The above form of the velocity gives also the correct asymptotic behavior: for going to infinity, being the area of the vortex. It is also important to notice that once the splitting is inserted into the integral at the right-hand side of (2.9), the original formulation (1.2) is recovered.

As a first sample case for the use of (2.9), consider an elliptical vortex having center of vorticity on the origin and semiaxes along and () along (the related quantities , , , and will be also used below). By using the angle , the curve is parametrized as , which is rewritten in terms of as Equation (2.10) defines a conformal map between the plane of the unit circle () and the physical one (). Due to the fact that the ellipse is a simple curve, the equation in cannot have another solution on unless Indeed, it also possesses the solution (notice that ). The solutions and are expressed in terms of in the following way: If the point belongs to the vortex, then and the same holds also for , while for , lies outside the unit circle and By conjugating both sides of (2.10) for , the Schwarz function follows naturally as the function of : The velocity is evaluated by rewriting (2.9) in the transformed plane : If , then both poles and are internal to and the above equation gives while if , lies outside the unit circle and , so that the velocity becomes However, in the present case the splitting of the Schwarz function (2.12) can be easily performed: which still gives the fields (2.14) and (2.15) through the relation (1.2). As discussed above, when the functional form of the Schwarz function is rather simple, as in (2.12), the old formulation (1.2) is convenient with respect to the new one (2.9). This is not the case of the class of vortices investigated in the present paper.

3. The Schwarz Function with Two Pole Singularities

The present paper deals with a uniform vortex having Schwarz function of its boundary with two pole singularities (it is worth noticing that the vortex having Schwarz function of its boundary with only the pole : is a circular, with center in and radius ) on a suitable transformed plane, with the aim to investigate the corresponding self-induced velocity field. For , the Schwarz function is considered. The residues , in (3.1) are assumed to be nonvanishing complex numbers (notice that one of them, e.g. , can be assumed unitary without loss of generality), while the poles , are chosen outside and the origin (). It is also assumed that their conjugates with respect to , that is, and , satisfy the two conditions and The Schwarz function (3.1) gives the position in the point as The point given by (3.2) can move counterclockwise or clockwise along , even if runs always counterclockwise on In order to specify the direction in which moves on , a parameter is introduced, which holds when runs counterclockwise and in the other case. It will be evaluated in Section 4.

Analytic continuations of the Schwarz function (3.1) and of the map (3.2) outside will be considered in the following, so that the definition of the inverse map outside the vortex boundary needs also to be discussed (see Section 6). It is shown that exists almost everywhere, unless on a closed curve , the inside of which has a vanishing area (see Figure 2).

Figure 2: Sketch of the invariant circle in the planes of the variables and and its image in the physical plane. The points and their images are represented with green dots. The same colours (blue and red) are used for the arcs that are images through the maps and of the two arcs in which is divided by the points For running on , the point moves on the closed curve , first in a direction and then in the reverse one. is composed by two superimposed arcs of the circle (yellow line) between the points and : its inside has a vanishing area.

3.1. Constants , , and

In the following, the poles and the corresponding residues of the Schwarz function (3.1) will often appear combined into the constants: moreover the related quantities: , , and will be also employed. In terms of the above constants, for example, the compact form of a vector which is tangent to the vortex boundary on its point : is obtained. The function (3.4) suggests a first constraint on the Schwarz one (3.1): cannot vanish on The nonvanishing zeros of are given by , with being the branch of the square root of for which (see also Appendix D. Notice that and its modulus verifies the relation , with being the phase difference ). As a consequence, in order to enforce on the unit circle, the points are hereafter assumed external to Other important constraints will be discussed in the next section.

3.2. Map and Constraints on Schwarz Function

The Schwarz function (3.1) cannot be assigned in an arbitrary way: for the curve is the boundary of a uniform vortex, it must be simple. This constraint can be enforced by requiring that for any , the equation in has no other solutions on , unless The only solution of that equation which differs from is so that the above condition results to be equivalent to the one: for each Equation (3.5) implies also that the points and go on the same point : the analytic continuation of the function (3.2) cannot be defined on the whole -plane, but only on a suitable subset of it including (see Section 5).

It is worth noticing here that the function (3.5) maps the points and (on which goes to the infinity) one in the other one and viceversa: for this reason these points and will be called hereafter as “conjugate” through the map Moreover, goes in and viceversa is mapped in ; the same occurs for , which goes in and viceversa is transformed in More details about the map can be found in Appendix A, in particular the “viceversa” parts of the above statements are trivial consequences of the property (A.1). As well known and summarized also in that appendix, transforms any circle in another circle. In the important example of the unit circle, its image is for the circle having center and radius given by the following relations: while becomes a straight line in the case of As discussed below, the position of relative to will be one of the key-points in order to understand the analytical structure of the velocity (see Section 4).

Coming back to the constraint on , the condition with leads from the definition (3.5) to the following equation in : which has no solutions if and only if where If the constraint (3.8) is verified, cannot intersect : this fundamental property, implying that is a simple curve, depends only on the function (3.5), or on the constants , , and (3.3). Hereafter, it will be assumed always satisfied.

3.3. Invariant Circles for the Map

An interesting issue about the map (3.5) lies in searching a circle which is transformed in itself by : such a circle will be called invariant. Two families at one parameter () of invariant circles have been found. In the first (and most important) one, center and radius, indicated with , in the -plane and with , in the -plane, are the following functions of : (the branch of the square root affects the sign of , but not the definition of the center). Centers (3.9) move along a straight line parallel to and passing on the point for running on the real axis. The importance of the circles of the first family lies in the fact that one of them (called hereafter, see Figure 2) will be used in order to define the inverse map It divides the complex plane in two parts which are mapped one in the other by the function (3.5), one of them being the image of the physical plane. As shown in Figure 2, another interesting feature of is that its image , called , results from the overlapping of two equal arcs of a circle (see Appendix E for details): for running on , moves along first in a direction and then in the reverse one. The branch of the inverse map, as well as the geometry of the two arcs on (unless their endpoints, which do not change with ), will depend on the value of the parameter (see Appendix D).

Centers of the second family lie along the straight line orthogonal to the one of the first family, but still passing through the point Center and radius of a circle are in that case the functions of : being larger than An inspection of the formulae (3.9) and (3.10) shows that circles of the first family intersect on the points Notice also that is divided into two complementary arcs by the points : if the point runs on from to in the counterclockwise sense, its image moves on the same circle and between the same points, but in the reverse direction. Finally, it is found that the function maps the points in the endpoints of the two arcs of which form

4. Vortex Classification

A classification of the vortices having the Schwarz function of their boundaries of the form (3.1) will be proposed below. It is based on two important properties of the maps (3.5) and (3.2) and allows us to build the inverse map and then the analytical velocity field.

The first property specifies the relative position of the curve with respect to the unit circle (hereafter, the case , in which the circle becomes a straight line, is excluded): can be internal to (the vortex is classified of kind ), or external and including (kind ), or external and not including (kind ). Vortices of the first kind have the inside of , say , naturally decomposed as joined with an annular set external to In order to use compact notations, this annular set or the whole (for vortices of kind or ) will be hereafter indicated with In turn, vortices of the second kind have the inside of naturally decomposed as joined with an annular subset external to , that will be indicated with For vortices of kind , the same symbol will indicate the unbounded region external to both circles and Appendix B is devoted to a comprehensive discussion of the first property, the results of which are summarized in Table 1. The above classification enables us to specify where goes through the function : it is found (see Appendix C for details) that is mapped onto itself for vortices of kind , while it goes onto the outside or the inside of for vortices of kind or , respectively.

Table 1: Behaviour with respect to the first property, consequences on the position of the circle and corresponding constraints on , and The quantities and are defined in (B.1).

A second property specifies where goes via the other map : it can go onto the inside of the vortex (which is classified of kind ) or onto its outside (kind ), depending on the values of the constants , and on the positions of the poles and with respect to the unit circle. This issue is investigated by evaluating if the image of the origin in the -plane (which is still the origin in the physical one) lies or not inside To this aim, the logarithmic index of the point with respect to the curve is considered. The residues (times ) of the integral (4.1) on the poles and (if they lie in ) hold , while those on and (if they lie in ) hold (notice that both pairs of points are conjugate through the map ). Results of the discussion of the logarithmic index (4.1) are summarized in Table 2.

Table 2: Behaviour with respect to the second property, consequences on the map and corresponding constraints on the poles ,

In our classification, the vortices belonging to a given class are of the same kind with respect to both properties: a class is identified by the couple of numbers which indicate the kinds with respect to the first and second properties. As an example, vortices in the class have the circle external and not including and is the outside of the vortex.

The last issue about the map concerns the orientation of the vortex boundary, that is, the evaluation of To this aim, the logarithmic index of the point with respect to has to be evaluated: it holds when is oriented counterclockwise and in the other case. In the last integral, the residue (multiplied times ) on the point holds , the ones on and (if they lie in ) give and the residue on (if it lies inside ) holds By accounting for the behaviour with respect to the second property summarized in Table 2, it can be easily shown that any vortex of kind has a counterclockwise () oriented boundary, while the orientation of the boundary of any vortex of kind is clockwise ().

5. Geometrical Discussion

In the present section, the information about the maps and given in Sections 3 and 4 are joined to give a comprehensive picture of their geometrical properties. In order to reach an intuitive representation of the behaviour of these functions, in Figures 3, 4, and 5 families of circles will be transformed through these two maps. Hence, the circles and their images are drawn with the same colour.

Figure 3: Planes of the variables: (first column), (second), and (third) for vortices of kind with respect to the first property. In the first row, a vortex of kind (, , , ) and in the second one a vortex of kind (, , , ) are investigated. Two families of circles are considered in the -plane: one inside (yellow lines) and the other one outside