Abstract
We set up general equations of motion for point vortex systems on closed Riemannian surfaces, allowing for the case that the sum of vorticities is not zero and there hence must be counter-vorticity present. The dynamics of global circulations which is coupled to the dynamics of the vortices is carefully taken into account. Much emphasis is put to the study of vortex pairs, having the Kimura conjecture in focus. This says that vortex pairs move, in the dipole limit, along geodesic curves, and proofs for it have previously been given by S. Boatto and J. Koiller by using Gaussian geodesic coordinates. In the present paper, we reach the same conclusion by following a slightly different route, leading directly to the geodesic equation with a reparametrized time variable. In a final section, we explain how vortex motion in planar domains can be seen as a special case of vortex motion on closed surfaces.
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1 Introduction
This paper extends previous results in Gustafsson (2019) on multiple point vortex motion on closed Riemannian surfaces of arbitrary genus to cases in which the sum of the vorticities is not zero, and therefore a counter-vorticity must be present. This occurs for example when there is only one-point vortex. For reasons dictated by a Hodge decomposition, the counter-vorticity is naturally chosen to be uniformly distributed on the surface.
Special for our investigations is that we, in the case of higher genus, carefully take the circulation around the holes into account and make sure that the Kelvin law of preservation of circulations is satisfied. This was done already in Gustafsson (2019), but in the case of a counter-vorticity being present one has to take some extra steps. It is natural to choose fixed closed curves on the surface to make up a basis for the first homology group, but the circulation around such curves will in general not be conserved in time. We handle the so arising problem by considering these circulations as free variables in the phase space, in addition to the locations of the vortices. If there are n vortices and the genus of the surface is \(\texttt {g}\), the phase space will thus have real dimension \(2n+2\texttt {g}\).
The Hamiltonian function will, as usual, be a renormalized kinetic energy for the flow, see (5.28), but in the presence of a counter-vorticity the symplectic form has to be accordingly adapted, see (6.1). With this done the dynamical equations come out to be the expected ones (Theorem 6.1). After having set up the vortex dynamics in general (Sects. 5 and 6) and after a short discussion of the single vortex case in genus zero (Sect. 7), we turn in Sect. 8 to the question of motion of vortex pairs, i.e., systems consisting of two point vortices of equal strength but rotating in the opposite direction. In this case, there is no counter-vorticity. The conjecture of Kimura (1999), saying that a vortex pair in the dipole limit moves along a geodesic curve, has been a major source of inspiration for the present paper. The same applies for papers (Koiller and Boatto 2009; Boatto and Koiller 2015) by S. Boatto, J. Koiller. These latter papers actually contain a proof of Kimura’s conjecture, and in the present paper we try to clarify the situation further by connecting the dipole trajectory directly to the geodesic equation associated with the metric.
Throughout this paper, we think of a Riemannian surface (a Riemannian manifold of dimension two) as a Riemann surface provided with a metric which is compatible with the conformal structure, and hence can be written on the form
We are then led into more or less classical function theory on Riemann surfaces. Our treatment is based entirely on local holomorphic coordinates. For example, the logarithmic singularities in the stream function will look like, up to a constant factor,
near a point vortex located at w. Here, the constant term \(c_0(w)\) is a kind of “connection” (0-connection, up to a sign, in our terminology). In potential theory, it is known as a capacity function, or “Robin function” if \(\psi \) is viewed as a Green function, and in fluid dynamics it goes under the name “Routh’s stream function,” a kind of stream function for the motion of the vortex. It has an inhomogeneous transformation law, which can be translated into saying that
is a conformally invariant Riemannian metric (usually not identical with (1.1) though).
The corresponding flow field \(\nu = ({\partial \psi }/{\partial y})\mathrm{d}x-({\partial \psi }/{\partial x})\mathrm{d}y=-*\mathrm{d}\psi \), considered here as a one-form or a covariant vector field, looks like
near the vortex point. The term named “complex conjugates” is inserted to make possible for \(\nu \) to be real-valued. Also in this expansion, the constant term is a “connection.” More precisely,
is an affine connection (or 1-connection in our terminology) in the sense of differential geometry. It appears in expressions for covariant derivatives, and after subtraction of the affine connection
derived from the metric (1.1) it gives the speed of the vortex. Indeed, citing from Gustafsson (2019) (Lemma 7.1 there) this speed is given by
where \(\Gamma \) is the strength of the vortex. This formula is also implicitly contained in Theorem 6.1 in the present paper.
One step further, it is possible to proceed directly to dipole flow by differentiating with respect to w:
Again the constant term defines a connection. To be precise, \(q(w)=-6c_2(w)\) is a projective, or Schwarzian, connection (or 2-connection); see Sect. 4, as well as relevant parts of Gustafsson (2019). This term gives some pieces of information on the fine structure of the flow near the dipole; however, it seems not to be directly related to the speed of the dipole. The differential \(\mathrm{d}w\) is to contain information about the orientation of the dipole and can be used to specify in which direction it is going to move. Indeed, all evidence shows that the dipole moves with infinite speed along the geodesic directed perpendicular to its orientation. This is Kimura’s conjecture, which was confirmed in Koiller and Boatto (2009) and Boatto and Koiller (2015) and which we make attempts to further clarify in the present paper.
Specifically, we show that the equations of motion for a vortex pair in a suitably scaled dipole limit reduce to the geodesic equation for the center w of the dipole. We write this geodesic equation on the form
with t being an arbitrary parameter, for example, Euclidean arc length. See Theorem 8.1.
The Kimura conjecture and other matters related to dipole motion have been discussed also in Smith (2011), Smith and Nagem (2013), Rodrigues et al. (2018), Koiller (2020), Krishnamurthy et al. (2021), and from slightly different points of view in Chorin (1973), Kulik et al. (2010), Holm and Jacobs (2017), and Cawte et al. (2019). It may be remarked that Kimura’s conjecture is counterintuitive. The reason to think so is that vortex motion in general is governed by global laws on the manifold, like the structure of Green functions and other harmonic functions, whereas the geometry of geodesics is an entirely local matter. If the geometry of the manifold is changed at one place, then this will not affect what geodesics look like at another place. It will, however, change the structure of Green functions and the general vortex dynamics everywhere. The solution of this paradox is that vortex dipoles are highly singular. A vortex pair is partly governed by the global harmonic structure, but in the limit, when the vortex pair becomes a dipole, this harmonic structure is completely overruled by the differential geometric structure. In that limit, all terms in the dynamical equations containing harmonic functions become negligible compared to those terms depending on the metric only. This fact shows up in the two dynamical equations above: The motion (1.2) of a single vortex is governed by \(r_{\mathrm{metric}}(w)\) and \(r_{\mathrm{robin}}(w)\) to equal parts, whereas only \(r_{\mathrm{metric}}(w)\) appears in equation (1.3) for a dipole. See Sect. 8 for further details.
In the final section of the paper, we discuss how point vortex motion in planar domains can be considered as a special case of vortex motion on surfaces. This is done by doubling the domain to a compact Riemann surface (the Schottky double), which is a standard tool as far as the conformal structure is concerned. What is special in our case is that we have to take the metric structure into account, and this becomes non-smooth in the doubling procedure. An interesting observation is that the boundary of the domain becomes a geodesic curve with respect to the natural planar metric (on each of the halves) of the Schottky double.
In general, the present paper is, in addition to the papers by S. Boatto and J. Koiller already mentioned, much in spirit of papers (Smith 2011; Smith and Nagem 2013; Ragazzo 2017; Ragazzo and Viglioni 2017; Wang 2021) by S. G. Llewellyn Smith, R. J. Nagem, C. Grotta Ragazzo, H. Viglioni, and Q. Wang. The paper (Bogatskiy 2019) by A. Bogatskiy contains ideas concerning the higher genus case which are somewhat similar to those in the present paper; see in particular Section A.3 in Bogatskiy (2019). We wish to mention also the work (Borisov and Mamaev 2006) by A. V. Borisov and I. S. Mamaev, and the very early article (Fridman and Polubarinova 1928) (cited in Borisov and Mamaev 2006; Smith and Nagem 2013) by A. A. Fridman and P. Ya. Polubarinova. That work, from 1928, may be one of the first papers on motion of dipoles and other higher singularities.
2 Fluid Dynamics on a Riemannian Surface
We consider the dynamics of a non-viscous incompressible fluid on a compact Riemannian manifold of dimension two. We follow the treatments in Schutz (1980), Arnold and Khesin (1998), and Frankel (2012) in the respects of treating the fluid velocity field as a one-form and in extensively using the Lie derivative. These sources also provide the standard notions and notations for differential geometry to be used. Other treatises in fluid dynamics, suitable for our purposes, are Marchioro and Pulvirenti (1994) and Newton (2001).
Traditionally, non-viscous fluid dynamics is discussed in terms of the fluid velocity vector field \(\mathbf{v}\), the density \(\rho \), and the pressure p of the fluid. The basic equations are the equation of continuity (expressing conservation of mass) and Euler’s equation (conservation of momentum):
These are to be combined with a constitutive law giving a relationship between p and \(\rho \). We shall take that to be the simplest possible: \(\rho =1\). Thus, \(\rho \) disappears from discussion, and the equation of continuity becomes
One can get rid also of p, because when \(\rho \) is constant, p appears only as a scalar potential in Euler’s equation, this equation effectively saying that
where this “something” \(=p\) afterward can be recovered, up to a (time dependent) constant.
So everything is extremely simple (in theory), and even more so in two dimensions. As is well known, every (oriented) Riemannian manifold of dimensions two can be made into a Riemann surface by choosing isothermal local coordinates \((x,y)=(x^1,x^2)\), by which \(z=x+\mathrm {i}y\) becomes a holomorphic coordinate and the metric takes the form
with \(\lambda >0\). Thus, the metric tensor \(g_{ij}\), as appearing in general in \(\mathrm{d}s^2=g_{ij}\mathrm{d}x^i \mathrm{d}x^j\), becomes \(g_{ij}=\lambda ^2\delta _{ij}\) in these coordinates. It turns out to be convenient to work with the fluid velocity 1-form (or covariant vector)
rather than with the corresponding vector field, which then becomes
The Hodge star operator takes 0-forms into 2-forms and vice versa and takes 1-forms to 1-forms. It acts on basic differential forms as
Thus, for 1-forms,
which can be interpreted as a rotation by ninety degrees of the corresponding vector.
In addition to the Hodge star, it is useful to introduce the Lie derivative \({\mathcal {L}}_\mathbf{v}\) of a vector field \(\mathbf{v}\). When acting on differential forms, this is related to interior derivation (or “contraction”) \(i(\mathbf{v})\) and exterior derivation d by the homotopy formula
The Hodge star and interior derivation are related by
where \(\mathbf{v}\) and \(\nu \) are linked via the metric tensor as above. Thus,
See Frankel (2012) for this identity, and for further details in general. We conclude that (2.3) is equivalent to the statement that \(*\nu \) is a closed form:
Locally, we can therefore write
for some (locally defined) stream function \(\psi \). The vorticity 2-form for a fluid is, in terms of the flow 1-form \(\nu ,\)
The Euler equation (2.2) can be written
Note that the left member involves the fluid velocity both as a vector field and as a form. The combination \(\frac{\partial }{\partial t}+{\mathcal {L}}_\mathbf{v}\) can be viewed as a counterpart, for forms, to the more traditional “convective derivative” (or material derivative) \(\frac{\partial }{\partial t}+\mathbf{{v}}\cdot \nabla \), implicitly used in (2.4), for vector fields. However, the two derivatives are not the same. The equivalence between (2.11) and (2.2) (when \(\rho =1\)) is a consequence of the identity
which can be directly verified by coordinate computations. See Schutz (1980), Frankel (2012) for details.
Since the pressure p does not appear in any other equation, the Euler equation on the form (2.11) only expresses, in analogy with (2.4), that the left member is an exact 1-form:
From the scalar \(\phi \) the pressure p then can be recovered via
On acting by d on (2.12), the local form of Helmholtz–Kirchhoff–Kelvin law of conservation of vorticity follows:
Since the right member of (2.12) is not only a closed differential form, but even an exact form, a stronger, global, version of the Helmholtz–Kirchhoff–Kelvin law actually follows. This can be expressed by saying that
for any closed curve \(\gamma (t)\) which moves with the fluid.
3 Green Functions and Harmonic Forms
3.1 The One-Point Green Function
In the sequel, M will be a closed (compact) Riemann surface provided with a Riemannian metric on the form (2.5).
Given a 2-form \(\omega \) on M, one can immediately obtain a corresponding Green potential \(G^\omega \) by Hodge decomposition, i.e., by orthogonal decomposition in the Hilbert space of square integrable 2-forms. The inner product for such forms is
The given 2-form \(\omega \) then decomposes as the orthogonal decomposition of an exact form and a harmonic form:
This Hodge decomposition can more precisely be written as
where \(G^\omega \) is normalized to be orthogonal to all harmonic 2-forms, namely satisfying
The constant in (3.2) necessarily equals the mean value of \(\omega \),
where V denotes the total volume (=area) of M:
When \(\omega \) is exact, as in (2.10), the second term in the right member of (3.2) disappears, hence
in this case. In the other extreme, when \(\omega \) is harmonic, i.e., is a constant multiple of \(\mathrm{vol}\), the first term disappears. Indeed, the whole Green function disappears:
For 1-forms, the inner product has the same expression as for 2-forms:
If \(\nu \) represents a fluid velocity, then \((\nu , \nu )_1\) is the kinetic energy of the flow (up to a factor). For a function (potential) u, we consider the Dirichlet integral \((\mathrm{d}u,\mathrm{d}u)_1\) to be its energy. Thus, constant functions have no energy.
The energy \({\mathcal {E}}(\omega ,\omega )\) of any 2-form \(\omega \) is defined to be the energy of its Green potential \(G^\omega \). Thus, for the mutual energy,
It follows, from (3.3) for example, that the volume form has no energy:
It was tacitly assumed above that the forms under discussion belong to the \(L^2\)-space defined by the inner product. However, the mutual energy sometimes extends to circumstances in which source distributions have infinite energy. This applies in particular to the Dirac current \(\delta _a\), which we consider as a 2-form with distributional coefficient, namely defined by the property that
for every smooth function \(\varphi \). Certainly \(\delta _a\) has infinite energy, but if \(a\ne b\), then \({\mathcal {E}}(\delta _a, \delta _b)\) is still finite and has a natural interpretation: It is the (one-point) Green function, or “monopole” Green function:
Here, the first equality can be taken as a definition, and then the second equality follows on using (3.2) and (3.3):
This shows in addition that G(a, b) is symmetric.
Changing now notations from a, b to z, w, where later w will have the role of being the location of a point vortex, the Green function G(z, w), as a function of z, has just one pole (at \(z=w\)). Its Laplacian, as a 2-form, is then
It is interesting to notice that among the two terms in the right member of (3.10), one has infinite energy and one has zero energy (\({\mathcal {E}}(\delta _w, \delta _w)=+\infty \), \({\mathcal {E}}(\mathrm{vol},\mathrm{vol})=0\)).
Remark 3.1
It is more common to treat the Dirac delta and the Laplacian (denoted \(\Delta \)) as “densities” with respect to the volume form. However, we find our usage convenient. In any case, the relationships are
Remark 3.2
(Notational remark) Letters like z, w will have the double roles of being complex-valued local coordinates on parts of the Riemann surface and of denoting the points on the surface for which the coordinates have the values in question. A more formal treatment could use, for example, \(P\in M\) for a point and \(z(P)\in {{\mathbb {C}}}\) for the corresponding coordinate value.
Remark 3.3
(Real versus complex notation) For future needs, we wish to clarify the relationship between real and complex coordinates in the context of tangent and cotangent vectors.
Let \(z=x+\mathrm {i}y\) be a complex coordinate on M and consider a curve \(t\mapsto z(t)\) in M (for example the trajectory of a vortex). Set \({\dot{z}}={\mathrm{d}z}/{\mathrm{d}t}\), and similarly for \({\dot{x}}\) and \({\dot{y}}\), so that \({\dot{z}}={\dot{x}}+\mathrm {i}{\dot{y}}\). The velocity of this moving point is primarily to be considered as a vector in the (real) tangent space of M (at the point under consideration). This gives, in the picture of viewing tangent vectors as derivations, the velocity vector
where it is understood that \(\dot{{\bar{z}}}=\overline{{\dot{z}}}\).
The real tangent space used above can in a next step be complexified, which means that one breaks the connection between \({\dot{z}}\) and \(\dot{{\bar{z}}}\) and consider them as independent complex variables. Equivalently, one allows \({\dot{x}}\) and \({\dot{y}}\) to be complex-valued. The so obtained complex tangent space can be decomposed as a direct sum of its holomorphic and anti-holomorphic subspaces, and then
becomes a natural and useful identification of the real tangent space with the holomorphic part of the complex tangent space. See Griffiths and Harris (1978), in particular Section 2 of Chapter 0, for further discussions. With this identification, \({\dot{z}}\) represents the velocity of z(t). Still one needs to keep in mind that \({\dot{z}}\) is just a complex number and that it is rather the preimage under \(\mathrm{proj}\) above that is the true velocity, as a real tangent vector.
The vector \(\mathbf{V}\) above corresponds, via the metric, to the covector
where \(\lambda =\lambda (x,y)=\lambda (z)\) (depending on context). Note that
as coefficients of \(\mathrm{d}z\) and \(\mathrm{d}{\bar{z}}\) (respectively) have similar roles (and signs) as the Wirtinger derivatives \(\partial /\partial z\), \(\partial /\partial {\bar{z}}\). For a covector \(\nu \) in general, we therefore define
so that
The corresponding (contravariant) vector is then, as in our fluid dynamical contexts,
3.2 Harmonic One-Forms and Period Relations
For later use, we record the formulas (differentiation with respect to z)
If \(\gamma \) is any closed oriented curve in M, then \(\oint _\gamma \mathrm{d}G(\cdot ,w)=0\), while the conjugate period defines a function
which, away from \(\gamma \), is harmonic in w and makes a unit additive jump as w crosses \(\gamma \) from the left to the right. The harmonicity of \(U_\gamma (w\)) is perhaps not obvious from outset since \(G(\cdot ,w)\) is not itself harmonic, but the deviation from harmonicity, namely the extra term in (3.10), is independent of w and therefore disappears under differentiation.
Differentiating (3.14) with respect to w gives
(integration with respect to z, Hodge star and d with respect to w). Adding and subtracting, we obtain the analytic (respectively anti-analytic) differentials
Let now \(\{\alpha _1, \dots , \alpha _\texttt {g}, \beta _1,\dots , \beta _\texttt {g}\}\) be representing cycles for a canonical homology basis for M such that each \(\beta _j\) intersects \(\alpha _j\) once from the right to the left and no other intersections occur (see Farkas and Kra 1992 for details). Then, there are corresponding harmonic differentials \(\mathrm{d}U_{\alpha _j}\), \(\mathrm{d}U_{\beta _j}\) obtained on choosing \(\gamma =\alpha _j,\beta _j\) in the above construction, and these constitute a basis of harmonic differentials associated with the chosen homology basis. Precisely, we have, for \(k,j=1,\dots ,\texttt {g}\),
The Kronecker deltas, \(\delta _{kj}\), come from the discontinuity (jump) properties of \(U_\gamma \) mentioned after (3.14).
The integrals of the basic harmonic differentials along non-closed curves become periods of two point Green functions:
Here, the integration in the left member is to be performed along a path that does not intersect the curve in the right member. These formulas are immediate from the definition (3.14) of \(U_\gamma \).
4 Affine Connections and Geodesic Curves
Besides differential forms, and tensor fields in general, certain kinds of connections with inhomogeneous transformation laws are relevant for point vortex motion. These arise naturally under “renormalization” processes, more precisely as the first regular terms in expansions of a polar singularities. The affine connections have the same meanings as in ordinary differential geometry, used to define covariant derivatives for example, and they play an important role in many areas of mathematical physics.
Some general references for the kind of connections we are going to consider are Schiffer and Hawley (1962), Gunning (1966, 1967, 1978), Gustafsson and Sebbar (2012). We define them in the simplest possible manner, namely as quantities defined in terms of local holomorphic coordinates and transforming in specified ways when changing from one coordinate to another.
Let \({\tilde{z}}=\varphi (z)\) represent a holomorphic local change of complex coordinate on a Riemann surface M and define three nonlinear differential expressions \(\{\cdot ,\cdot \}_k\), \(k=0,1,2\), by
The last expression is the Schwarzian derivative of \(\varphi \). For \(\{{\tilde{z}},z\}_0\), there is an additive indeterminacy of \(2\pi \mathrm {i}\), so actually only its real part, or exponential, is completely well defined. If z depends on w via an intermediate variable u, then the following chain rule holds:
Definition 4.1
An affine connection (or 1-connection) on M is an object which is represented by local differentials \(r(z)\mathrm{d}z\), \({\tilde{r}}({\tilde{z}})\mathrm{d}{\tilde{z}}\),...(one in each coordinate variable, and not necessarily holomorphic) glued together according to the rule
Definition 4.2
A projective connection (or Schwarzian connection, or 2-connection) on M consists of local quadratic differentials \(q(z)(\mathrm{d}z)^2\), \({\tilde{q}}({\tilde{z}})(\mathrm{d}{\tilde{z}})^2\), ..., glued together according to
One may also consider 0-connections, assumed here to be real-valued. Such a connection p(z) transforms according to
This means exactly that
is a Riemannian metric.
For a metric in general, \(\mathrm{d}s =\lambda (z){|\mathrm{d}z|}=e^{p(z)}|\mathrm{d}z|\), there is a natural affine connection \(r(z)=r(z)_{\mathrm{metric}}\) associated with it by
This can be identified with the Levi-Civita connection in general tensor analysis. The real and imaginary parts coincide (up to sign) with the components of the classical Christoffel symbols \(\Gamma _{ij}^k\). The Gaussian curvature of the metric is
The equation for geodesic curves \(z=z(t)\) is
where t is the arc length with respect to the metric. Written in another way,
The first version is just a reformulation of the usual equation in terms of Christoffel functions in ordinary differential geometry (see Frankel 2012), and it is also discussed in a complex analytic context in Gustafsson and Sebbar (2012).
The real part of (4.2) only contains information about the parametrization, while the imaginary part, namely
describes the geodesic curve in terms of an arbitrary real parameter t. The latter property is useful in the context of the motion of vortex pairs since the speed of such a pair tends to infinity as the distance between the two vortices goes zero, and time therefore has to be successively reparametrized.
5 Energy Renormalization and the Hamiltonian
5.1 The Renormalized Kinetic Energy
When \(\nu \) is the flow 1-form of an incompressible fluid, the equation of continuity says that \(d*\nu =0\) (see (2.8)). The vorticity 2-form is \(\omega =\mathrm{d}\nu \), and (3.5) holds. Thus, \(d(\nu +*\mathrm{d}G^\omega )=0\), and setting
it follows that \(\eta \) is harmonic: \(\mathrm{d}\eta =0=d*\eta \). Locally, we can write \(*\eta =\mathrm{d}\psi _0\) for some harmonic function \(\psi _0\). Then,
becomes a locally defined stream function for the flow, so that
Different local choices of \(\psi \) may differ by additive time dependent constants. The role of the additional term \(\psi _0\) in the stream function will be clarified when discussing the hydrodynamic Green function for planar vortex motion in Sect. 9.
The relation (5.1), when written on the form \(\nu =\eta -*\mathrm{d}G^\omega \), is an orthogonal decomposition of the flow 1-form \(\nu \) with respect to the inner product (3.7). Indeed,
since \(\eta \) is harmonic. It follows that (twice) the total (kinetic) energy \((\nu ,\nu )_1\) of the flow is given by
We shall now specialize on the point vortex case, having point vortices of strengths \(\Gamma _k\) located at points \(w_k\in M\) (\(k=1,\dots , n\)). However, we shall not assume that these strengths add up to zero. Hence, the sum
may be nonzero and there will then be a compensating uniform counter-vorticity. The total vorticity \(\omega =\mathrm{d}\nu \), which satisfies \(\int _M \omega =0\), appears as the right member in
Explicitly, we have (recall (3.6))
On using (3.13), the conjugate \(\alpha \)-periods of \(G^\omega \) can be expressed as
By differentiation and using also (3.15) and (3.16) (with \(\gamma =\alpha _j\)), we find that
Here, \(U^*_{\alpha _j}\) denotes a harmonic conjugate of \(U_{\alpha _j}\), whereby \(*\mathrm{d}U_{\alpha _j}=d(U^*_{\alpha _j})\). Similar relations hold for periods around \(\beta _j\), and for \({\bar{w}}_k\) derivatives.
The expression for the kinetic energy can in the point vortex case be written, at least formally,
However, the presence of the terms \(\Gamma _k^2G(w_k,w_k)\) makes the first term become infinite. In order to renormalize this singular behavior, we isolate the logarithmic pole in the Green function by writing
and expand, for a fixed w, the regular part in a power series in z as
We note from (5.10) that
Thus, the symmetry of H(z, w) gives
In Gustafsson (2019), the behavior of the coefficients in the expansion (5.10) under conformal mapping was discussed. For example, the coefficient \(h_{11}(w)\) transforms as the density of a metric, and it is indeed proportional to the given metric:
The coefficients \(h_0(w)\), \(h_1(w)\) and \(h_2(w)\) are subject to inhomogeneous transform laws, expressing that they, in certain combinations, behave as connections under conformal mappings.
For \(h_0(w)\), the law is
and this means that its exponential defines a metric, namely the Robin metric via
Metrics of this kind are implicit in the theory of capacity functions, as exposed in Sario and Oikawa (1969). It should be pointed out that our version of the Robin metric depends on the given metric, since the Green function itself depends on it. The Robin metric can also be adapted to given global circulations, and then it becomes more intrinsically hydrodynamic in nature. The coefficient \(h_0(w)\) is one example of a (coordinate) Robin function.
Now, letting w have the role of being a vortex point indicates that one could renormalize the kinetic energy by simply discarding the singular term \(\log |z-w|\), as this seems at first sight to produce just a circular symmetric flow, not affecting the speed of the vortex. However, this is not fully correct in the case of curved surfaces. The term \(\log |z-w|\) cannot be just removed; it needs be replaced by a term which counteracts the above inhomogeneous transformation law of \(h_0(w)\). Such a term comes naturally from the given metric \(\mathrm{d}s=\lambda (w)|\mathrm{d}w|=e^{\log \lambda (w)}|\mathrm{d}w|\). We see that minus \(\log \lambda (w)\) has the right properties, and it combines with \(h_0(w)\) into
This is indeed a function, the Robin function, and it appears naturally when writing the singularity of the Green function in terms of the distance with respect to the given Riemannian metric:
As for the infinite kinetic energy, we conclude that it should be renormalized by replacing the diagonal terms \(G(w_k, w_k)\) by \(R_{\mathrm{robin}}(w_k)\). This gives the same equations of motion as more direct approaches, or those available in the literature, like (Hally 1980; Boatto and Koiller 2015; Dritschel and Boatto 2015). Thus, the renormalized energy is
This depends on the locations \(w_1, \dots , w_n\) of the vortices (even the last term depends on these, although somewhat more indirectly).
Many authors start out directly with the Robin function (5.15), but there are some advantages with exposing the two terms in it as individual quantities. One is that it clarifies the structure of the vortex motion equations by separating harmonic contributions, such as \(h_0(w_k)\) and \(G(w_k, w_j)\), from differential geometric contributions, like \(\log \lambda (w_k)\). The first category of terms can be classified as nonlocal, coming from solutions of elliptic partial differential equations on the entire surface, whereas the second category are purely local in nature. There is also the contribution from global circulations, represented by the final term in (5.16), and this may be considered to be truly global. So the vortex motion is in principle governed by a balance between different categories of contributions, local, nonlocal, and global in nature. As we shall see later, this balance is drastically changed in the more singular case of dipole motion (Sect. 8). Then, only the local terms survive.
We need to elaborate further the expression (5.1) and relate it to the global circulations. The 1-form \(\eta \) is harmonic and hence can be expanded as
The coefficients \(A_k\), \(B_k\) are in view of (3.18), (3.19) given by
The circulations of the total flow \(\nu =\eta -*\mathrm{d}G^\omega \) then become
We shall treat the circulations \(a_1,\dots ,a_{\texttt {g}}\), \(b_1,\dots ,b_{\texttt {g}}\) as free (independent) variables, along with the locations \(w_1,\dots , w_n\) of the vortices. These variables will be the coordinates of the phase space, and together they determine \(\eta \) and \(\nu \). The locations of the vortices are given by complex variables, in contrast to the circulations which are real.
If \(\gamma \) is a closed oriented curve in M, fixed in time and avoiding the point vortices, then according to the Euler equation (2.12), and in the notation used there, the circulation of \(\nu \) around \(\gamma \) changes with speed
Here, we used also (2.6), (2.7), and (5.4). It follows, in particular, on choosing \(\gamma =\alpha _k, \beta _k\), that
5.2 Matrix Formalism and the Hamiltonian
The period matrix (written here in block form)
is symmetric and positive definite (see Farkas and Kra 1992 in general). In particular, P and Q are themselves symmetric and positive definite. As for R and \(R^T\), we need to be explicit with what are row and column indices above: In all entries above, k is the row index and j the column index. Thus, for example, \(R_{kj}=\oint _{\beta _k}*\mathrm{d}U_{\alpha _j}\).
Next, we write
As mentioned, these two column matrices together define a basis of the harmonic forms. Another basis is provided by the corresponding Hodge starred column vectors \(*\mathrm{d}U_\alpha \), \(*\mathrm{d}U_\beta \), in similar matrix notation. The relation between the bases is
Lemma 5.1
The two bases \(\{ \mathrm{d}U_{\alpha }, \mathrm{d}U_{\beta }\}\) and \(\{ *\mathrm{d}U_{\alpha }, *\mathrm{d}U_{\beta }\}\) are related by
Proof
One simply checks that the two members in (5.21) have the same periods with respect to the homology basis \(\{\alpha _j, \beta _j\}\). \(\square \)
We arrange also the circulations of the flow \(\nu \) into column vectors:
briefly written as
Similarly for other vectors of circulations, for example
The \(\alpha \)-periods of the conjugated Green function were computed in (5.6), and from (5.7) and (5.8) it follows that
Taking into account also the dependence of \(a_1,\dots ,a_\texttt {g}\), and doing the same for B and the \(\beta \)-periods, gives the total differentials
In matrix notation (5.17) becomes
and the contribution from \(\eta \) to the kinetic energy is the quadratic form
The last equality is based on straightforward computations. Note that P, Q, R are fixed matrices, while A and B depend on \(w_1,\dots ,w_n\), \(a_1,\dots ,a_\texttt {g}\), \(b_1,\dots , b_\texttt {g}\) via (5.23), (5.24).
We now define the Hamiltonian function, \({\mathcal {H}}\), as the renormalized kinetic energy of the flow considered as a function of the locations of the point vortices and of the circulations:
We have adjusted with a factor two in front of \({\mathcal {H}}\) in order to conform with standard formulas.
The left member of (5.28) exhibits the dependence of \({\mathcal {H}}\) on the points \(w_1,\dots , w_n\), but considering the \(w_j\) as coordinates the dependence is not analytic. To exhibit analytic dependence, one may therefore write
6 Hamilton’s Equations
6.1 Phase Space and Symplectic Form
To formulate Hamilton’s equation, one needs, besides the Hamiltonian function itself, a phase space and a symplectic form on it. The phase space will in our case consist of all possible configurations of the vortices, collisions not allowed, together with all possible circulations of the flow around the curves in the homology basis. Thus, we take it to be
Here, the \(w_j\) shall be interpreted as points on M, but in most equations below they will refer to complex coordinates for such points. Compare Remark 3.2.
Assuming that \(\Gamma \ne 0\) (recall (5.3)) the symplectic form on \({\mathcal {F}}\) is taken to be
The last expression makes sense also if \(\Gamma =0\), because by (5.18), (5.19) the factors \(\mathrm{d}a_j/\Gamma \) and \(\mathrm{d}b_j/\Gamma \) remain finite as \(\Gamma \rightarrow 0\). Thus, in both expressions above, the last term shall simply be removed if \(\Gamma =0\).
Let
denote a generic tangent vector of \({\mathcal {F}}\) viewed as a derivation. As for the first term, recall the conventions in Remark 3.3. Hamilton’s equations in general say that
One main issue below (in Sect. 6.2) will be to verify that with our choices of phase space, symplectic form and Hamiltonian function, the equations (6.2) really produce the expected vortex dynamics.
To evaluate (6.2), we first compute the left member as
Explicitly, (6.2) therefore says that
Here, the partial derivatives in the right members are implicit in
Recall the expressions (5.25), (5.26) for A, B in terms of the phase space variables a, b (as column matrices) and \(w_1,\dots , w_n\).
In the partial derivatives of \(R_{\mathrm{robin}}(w)\) (see (5.15)), we single out the two affine connections
(see (5.12) for the last equality). We then have the following alternative expressions for (ingredients of) the first term in \(\mathrm{d}{\mathcal {H}}\):
6.2 Dynamical Equations
The following theorem now makes the Hamilton equations (6.2) explicit.
Theorem 6.1
The dynamical equations for a point vortex system as above are
Recall the column vectors (5.23), (5.24):
Proof
The equations are obtained by identifying the expression (6.3) for \(i(\xi )\Omega \) with the expression (6.6) for \(\mathrm{d}{\mathcal {H}}\). Thus, in (6.3), the k:th term in the first sum,
is to be identified with the corresponding parts in the right member of (6.6), namely the first two terms. Together with (6.9), this gives immediately the first two terms in the equation for \({\dot{w}}_k\).
The third term comes from the term
in equation (6.6). On using (5.21), this can be rewritten as
Identifying here the coefficient for \(\mathrm{d}{\bar{w}}_k\) with the corresponding coefficient in (6.10) gives
as desired.
Finally, the term
in (6.6) is to be identified with
in (6.3). This gives
as desired. \(\square \)
7 Motion of a Single Point Vortex
In the case of a single vortex, Theorem 6.1 simplifies a little. We may then denote the vortex point \(w_1\) simply by w, and the strength \(\Gamma _1\) agrees with the total vorticity \(\Gamma \) for the point vortices. If in addition \(\texttt {g}=0\) then everything simplifies considerable. There is only one free variable, the location \(w\in M\) for the vortex, and the dynamical equation for this is
The Hamiltonian and the symplectic form are
It follows that if (and only if) the two metrics
are identical, up to a constant factor, then the vortex will never move, whatever its initial position is. In Boatto and Koiller (2015) and Ragazzo and Viglioni (2017), this is referred to as \(\mathrm{d}s_{\mathrm{metric}}\) being a “steady vortex metric,” or being “hydrodynamically neutral.”
Example 7.1
An obvious example of a hydrodynamically neutral metric is that of a homogenous sphere. Indeed, for a sphere of radius one we have, in coordinates obtained by stereographic projection into the complex plane, well-known formulas such as
The Hamiltonian function is constant,
and there is no motion of the vortex.
8 Motion of a Vortex Pair in the Dipole Limit
For a vortex pair \(\{w_1, w_2\}\) with \(\Gamma _1=-\Gamma _2\), we have \(\Gamma =\Gamma _1+\Gamma _2=0\), and hence there is no compensating background vorticity. The circulations a and b will be time independent by the last equation in Theorem 6.1 and are not needed in phase space, which then simply becomes
with symplectic form
The Hamiltonian is the same quantity as before, see (5.28), but it may now be considered as a function only of \(w_1\) and \(w_2\). The circulations a and b are fixed parameters, given in advance.
However, the period vectors A and B still depend on time via \(w_1\) and \(w_2\). Indeed, using (3.20) and (3.21) we have
The Hamiltonian is given by
and the dynamics of the vortex pair \(\{w_1,w_2\}\) becomes, by (6.4),
In place of \(w_1\) and \(w_2\), one may turn to \(w=\frac{1}{2}(w_1+w_2)\) and \(u=\frac{1}{2}(w_1-w_2)\) as coordinates. These are similar to the “center-arrow coordinates” used in Boatto and Koiller (2015), Smith and Nagem (2013). Then,
We are interested in the dipole limit \(u\rightarrow 0\). More precisely, we shall let \(|u|\rightarrow 0\) while requiring \(\arg u\) to be stable and approach a definite limit. This limit then determines the orientation of the dipole.
In view of the symmetry \(H(w_1,w_2)=H(w_2,w_1)\), we have expansions
(coupled signs throughout). The latter equation uses the affine connection \(r(w)=r_{\mathrm{metric}}(w)\), see (6.7) or (4.1). We record also
Using these expansions we obtain, for the first matrix in the Hamiltonian (8.2),
When acting with \((\Gamma _1, -\Gamma _1)\) on both sides of the matrix (8.5), the last three terms in the final expression disappear and the result becomes, up to \({\mathcal {O}}(|u|^2)\),
For the full Hamiltonian (8.2), we therefore have, up to \({\mathcal {O}}(|u|^2)\),
In the above expression, the final term remains bounded as \(u\rightarrow 0\) and hence is negligible in this limit compared to the first term. Therefore, the leading terms in this limit are given by
This is essentially a constant factor times log of the distance (in the metric) between \(w_1=w+u\) and \(w_2=w-u\). Indeed, we recover the simple and beautiful formula
of Boatto and Koiller. See equations (24) and (25) in Boatto and Koiller (2015). Compare also Ragazzo and Viglioni (2017). The distance is taken with respect to the Riemannian metric. The error term \({\mathcal {O}}(1)\) can be identified with a what is called the “Batman function” in Boatto and Koiller (2015).
To proceed toward the Hamilton equations (6.2), we differentiate the leading term in (8.7) and insert the metric affine connection \(r(w)=r_{\mathrm{metric}}(w)\) as defined by (6.7),
Using (8.4), we expand the symplectic 2-form given by (8.1) in terms of w and u as
Interior derivation by
then gives, up to terms of order \({\mathcal {O}}(|u|^2)\),
Comparing with (8.8), we see that the dynamics of the vortex pair is described by the two equations
valid up to order \({\mathcal {O}}(|u|^2)\) as \(u\rightarrow 0\).
Equation (8.10) is used to eliminate \({\dot{w}}\) in (8.9), which then becomes
again up to \({\mathcal {O}}(|u|^2)\). Here, the left member cancels with one of the terms in the right member, and the rest can be written, after division by u,
now up to an error of order \({\mathcal {O}}(|u|)\). In this equation, the final main term in the right member is of a smaller magnitude than the first two terms and is swallowed by the allowed general error \({\mathcal {O}}(|u|)\). Thus, we arrive at
valid with an error of at most \({\mathcal {O}}(|u|)\) as \(u\rightarrow 0\).
Equation (8.11) essentially comes from (8.9), and it is to be combined again with (8.10). For the latter, it is enough to use the simplified form
which only uses the leading terms, and for which the error is still at most \({\mathcal {O}}(|u|)\). Inserting (8.12) in (8.11) results in the master equation
One problem with (8.13) is that the speed \(\mathrm{d}w/\mathrm{d}t\) becomes infinite, along with the first term, in the dipole limit. However, this problem only affects the real part of the equation. For the imaginary part, one can replace true time t by an arbitrary parameter, which may be scaled with u so that \(\mathrm{d}w/\mathrm{d}t\) remains finite as \(|u|\rightarrow 0\). Alternatively one may scale \(\Gamma _1\) with u so that the left member of (8.12) remains finite. Then, one can still think of any new parameter t as a time variable. In any case, we take imaginary parts of (8.13) and obtain
Equation (8.12) shows that the directions of u and \(\mathrm{d}w/\mathrm{d}t\) are related as
where the plus sign is to be chosen if \(\Gamma _1>0\), the minus sign if \(\Gamma _1<0\). Now (8.14) and (8.15) taken together give the final law for the motion of the center w of the vortex pair in the dipole limit:
As explained in Sect. 4 (see specifically equation (4.3)), (8.18) is exactly the equation for a geodesic curve when expressed in an arbitrary parameter t. One may notice that (8.14) can be written in the parameter-free form
(similarly for (8.16)), confirming the fact that the geometry of the dipole trajectory has a meaning independent of any choice of parameter.
The real part of (8.13) says, in view of (6.7), that
hence that \({|u|}{\lambda (w)}= C\) (constant) along each trajectory. By (8.12), this also gives
Thus, along each trajectory \(\lambda (w)\) is proportional to one over the velocity, and in this sense has the same role as the refraction index in optics.
We summarize the most essential parts of the above discussion as
Theorem 8.1
The dynamical equations for a vortex pair in the dipole limit are identical with the geodesic equation for the metric \({d}s=\lambda (w)|{d}w|\), namely
Here, \(w=w(t)\) is the location of the dipole, t is an arbitrarily scaled time parameter chosen such that \(\mathrm{d}w/\mathrm{d}t\) is finite.
Remark 8.1
It is possible to understand why dipole move along geodesics by thinking of vortex pair as a “wave front,” in an optical analogy. Equation (8.15) says that the motion is perpendicular to the wave front (the line segment from \(w-u\) to \(w+u\)). Equation (8.14) then expresses that if the front of a vortex pair is not aligned with the level line of \(\lambda (w)\) then the direction of u (representing the wave front) changes in such a way that the curve w(t) bends toward higher values of \(\lambda \).
Being slightly more direct and exact, on taking t to be Euclidean arc length the first term in (8.18) is the ordinary curvature for the curve traced out by w(t). The second term can be viewed as the inner product between the gradient of \(\lambda (w)\) (this gradient can be identified with \(\overline{r(w)}\)) and \(\mathrm{d}w/\mathrm{d}t\) rotated 90 degrees to the right. Letting \(\theta \) denote the angle between the gradient of \(\lambda (w)\) and the velocity vector \(\mathrm{d}w/\mathrm{d}t\), we can write
This is compatible with the laws of optics, for example Fermat’s law, and also with “Snell’s law” (see for example Guillemin and Sternberg 1984; Cawte et al. 2019) in the somewhat singular case that \(\lambda (w)\) jumps between two constant values.
9 Remarks on Vortex Motion in Planar Domains
Vortex motion in a planar domain can easily be treated as a special case of vortex motion on Riemann surfaces by turning to the Schottky double of the planar domain. For simplicity, we shall only discuss the case of one single vortex in the planar domain. The case of several vortices will be similar in an obvious way. The ideas in this section extend to cases of vortex motion on general open Riemannian surfaces with analytic boundary.
Let \(\Omega \subset {{\mathbb {C}}}\) be the planar domain, assumed to be bounded by finitely many real analytic curves. The Schottky double, first described in Schottky (1877), is the compact Riemann surface \(M=\hat{\Omega }\) obtained by completing \(\Omega \) with a “backside” \({\tilde{\Omega }}\), having the opposite conformal structure, and gluing the two along the common boundary. Thus, \(\hat{\Omega }=\Omega \cup \partial \Omega \cup {\tilde{\Omega }}\) in a set theoretic sense, and the conformal structure becomes smooth over \(\partial \Omega \), as can be seen from well-known reflection principles. If z is a point in \(\Omega \subset M\), then \({\tilde{z}}\) will denote the corresponding (reflected) point on \({\tilde{\Omega }}\subset M\). Both z and \({\tilde{z}}\) can also be considered as points in \({{\mathbb {C}}}\), then serving as coordinates of the mentioned points in M (holomorphic respectively anti-holomorphic coordinates), and as such they are the same: \(z={\tilde{z}}\in {{\mathbb {C}}}\).
In our case, we also need a Riemannian structure, with a metric. This is to be the Euclidean metric on each of \(\Omega \) and \({\tilde{\Omega }}\). But these do not fit smoothly across curved parts of \(\partial \Omega \), the match will only be Lipschitz continuous. However, that is good enough for our purposes because the vortex will anyway never approach the boundary. (In the case of several vortices, it is, however, possible to make up situations in which vortices do reach the boundary.)
The metric on \(M=\hat{\Omega }\) is thus to be
To see how this behaves across \(\partial \Omega \), we need a holomorphic coordinate defined in a full neighborhood of this curve in M. A natural candidate can be defined in terms of the Schwarz function S(z) for \(\partial \Omega \), a function which is defined by its properties of being holomorphic in a neighborhood of \(\partial \Omega \) in \({{\mathbb {C}}}\) and by satisfying
See Davis (1974) and Shapiro (1992) for details about S(z). We remark that \(z\mapsto \overline{S(z)}\) is the (local) anti-conformal reflection map in \(\partial \Omega \) and that \(S'(z)=T(z)^{-2}\), where T(z) is the positively oriented and holomorphically extended unit tangent vector on \(\partial \Omega \).
The complex coordinate z in \(\Omega \) extends, as a holomorphic function, to a full neighborhood of \(\Omega \cup \partial \Omega \), both when this neighborhood is considered as a subset of \({{\mathbb {C}}}\) and when it is considered as a subset of M. The first case is trivial, and the second case depends on \(\partial \Omega \) being analytic. In terms of the Schwarz function, this second extension is given by
In the latter expression, \(\overline{S({\tilde{z}})}\), \({\tilde{z}}\) is to be interpreted as a complex number. When the metric on M is expressed in the coordinate (9.3), it becomes
In the second case (with \({z}\in {{\mathbb {C}}}\setminus {\overline{\Omega }}\)), \(z=\overline{S({\tilde{z}})}\), \({\tilde{z}}\in {\tilde{\Omega }}\), whereby \({\tilde{z}}=\overline{S(z)}\) and so \(|\mathrm{d}{\tilde{z}}|=|S^\prime ({z})||\mathrm{d}{z}|\). Thus, (9.4) is consistent with (9.1). We see from the coordinate representation (9.4) that the metric is only Lipschitz continuous across \(\partial \Omega \). This is the best one can expect.
The associated affine connection (6.7) is in the coordinate (9.3) given by
where
See Sect. 4 for notations. Thus, r(z) is discontinuous across \(\partial \Omega \) and it is natural to represent it on this curve by its mean value:
Example 9.1
Let \(\Omega ={{\mathbb {D}}}\). Then, \(S(z)=1/{z}\) and the coordinate z in (9.3) extends to the entire complex plane, thus representing all of \(M={{\mathbb {D}}}\cup \partial {{\mathbb {D}}}\cup {\tilde{{{\mathbb {D}}}}}\) except for the point \({\tilde{0}}\in {\tilde{{{\mathbb {D}}}}}\). The metric expressed in this coordinate becomes
The affine connection similarly becomes
The geodesic curves in \(\Omega \) are of course the (Euclidean) straight lines (similarly in \({\tilde{\Omega }}\)), geodesic curves crossing \(\partial \Omega \) are straight lines reflecting into the other side under equal angles on \(\partial \Omega \) (just as ordinary optical reflection), while \(\partial \Omega \) is in itself a geodesic curve. The latter is intuitively obvious since at any point on \(\partial \Omega \) there should be one geodesic in the tangential direction, and this has no other way to go than to follow the boundary.
To confirm the last statement analytically, let t be an arc length (with respect to the Euclidean metric) parameter along \(\partial \Omega \), so that \(T(z)=\frac{\mathrm{d}z}{\mathrm{d}t}\) on \(\partial \Omega \). The curvature \(\kappa \) of \(\partial \Omega \) is
and using that \(T(z)\overline{T(z)}=1\) on \(\partial \Omega \) one finds that
In particular \(T'(z)\), and so \(r_{\mathrm{MV}}(z)T(z)\) (by (9.6)), is purely imaginary on \(\partial \Omega \). Combining with (9.6), it follows that
hence
Thus, the geodesic equation (8.18) holds for the curve \(\partial \Omega \), as claimed.
We remark that the curvature \(\kappa \) of the boundary curve \(\partial \Omega \) appears also in the expression for the Gaussian curvature for the metric on M. That curvature vanishes on \(\Omega \) and on \({\tilde{\Omega }}\), whereas on \(\partial \Omega \) it has a singular contribution with density \(2\kappa \) with respect arc-length measure on \(\partial \Omega \).
A single vortex in a planar domain \(\Omega \) moves along a level line of the appropriate Robin function, or Routh’s stream function Lin (1941a), Lin (1941b), Lin (1943)). If the vortex is near the boundary, then it follows the boundary closely, with high speed. From the perspective of the Schottky double, the boundary conditions for planar fluid motion are such that there is automatically a mirror vortex on the other side in the double; thus, we really have a vortex pair close to \(\partial \Omega \) on the double. In the limit, this becomes a vortex dipole, moving with infinite speed along the geodesic \(\partial \Omega \).
Considering in some more detail such a symmetric vortex pair, with vortex locations \(w\in \Omega \) and \({\tilde{w}}\in {{\tilde{\Omega }}}\), we first notice that the Green function \(G^\omega (z)\) for \(\omega =\delta _w-\delta _{{\tilde{w}}}\) simply is the antisymmetric extension to the Schottky double of the ordinary Green function \(G_\Omega (z,w)\) for \(\Omega \):
Then, the stream function \(\psi \) in (5.2) becomes what is sometimes called the hydrodynamic Green function, which depends on the prescribed periods. This function, which can be traced back (at least in special cases) to Koebe (1916), Lin (1943), has more recently been discussed in for example Cohn (1980), Flucher and Gustafsson (1979), Flucher (1999), and Gustafsson and Sebbar (2012).
We wish to clarify how this hydrodynamic Green function is related to the modification, in the beginning of Sect. 5, of the general Green function flow \(-*\mathrm{d}G^\omega \) by an additional term \(\eta \). To do so we fix, in the case of a Schottky double \(M={\hat{\Omega }}\), the homology basis in such a way that the curves \(\beta _j\), \(j=1,\dots ,\texttt {g}\), closely follow the inner components of \(\partial \Omega \), and each curve \(\alpha _j\) goes from the j:th inner component of \(\partial \Omega \) through \(\Omega \) to the outer component, and then back again on the backside.
We also introduce the harmonic measures \(u_j\), \(j=1,\dots ,\texttt {g}\), here defined to be those harmonic functions in \(\Omega \) which takes the boundary value one on the designated (number j) inner component of \(\partial \Omega \) and vanishes on the rest of \(\partial \Omega \). Their differentials \(\mathrm{d}u_j\) extend harmonically to the Schottky double with \(\oint _{\alpha _k} \mathrm{d}u_j=-2\delta _{kj}\), \(\oint _{\beta _k} \mathrm{d}u_j=0\). Thus, \(\mathrm{d}u_j=-2\mathrm{d}U_{\beta _j}\) in terms of our general notations (as in (3.18) and (3.19)).
In the block matrix notation of (5.21), we have
where \(P=(P_{kj})\), \(R=(R_{kj})\) and (see (3.18), (3.19))
The last integral can be written
and it is easy to see that it is zero because of canceling contributions from \(\Omega \) and \({\tilde{\Omega }}\) due to the symmetry of \(\mathrm{d}u_k\) and \(\alpha _j\) going the opposite way on the backside.
As a general conclusion, we therefore have that \(R=0\) in the matrix (5.21) when M is the Schottky double of a planar domain. As a consequence,
Similarly, the other equation contained in (5.21) becomes
Turning now to the flow \(\eta \) in (5.17), this is necessarily symmetric on \(M={\hat{\Omega }}\), hence
It follows that the coefficients \(A_j\) in (5.17) vanish, whereby that equation becomes
In terms of the stream function \(\psi =G^\omega +\psi _0\) (recall (5.2)) this gives, on inserting also (9.7),
with \(C_j=\sum _{i=1}^{\texttt {g}} B_i Q_{ij}\). It follows in particular that \(\psi _0\), and hence all of \(\psi \), is single-valued on \(\Omega \).
In summary, the total stream function is
and it is single-valued when restricted to \(\Omega \). It is clear that the \(C_j\) will actually depend on w, and for symmetry reasons the above formula eventually takes the well-known form
for some constants \(C_{ij}\) subject to \(C_{ij}={C_{ji}}\).
Data Availability
All data needed are contained in the manuscript.
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Gustafsson, B. Vortex Pairs and Dipoles on Closed Surfaces. J Nonlinear Sci 32, 62 (2022). https://doi.org/10.1007/s00332-022-09822-9
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DOI: https://doi.org/10.1007/s00332-022-09822-9
Keywords
- Point vortex
- Vortex pair
- Vortex dipole
- Geodesic curve
- Affine connection
- Projective connection
- Robin function
- Hamiltonian function
- Symplectic form
- Green function