Vortex pairs and dipoles on closed surfaces

We consider the dynamics of a non-viscous incompressible fluid on a compact Riemannian manifold of dimension two. We follow the treatments in [46, 1, 15] 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 [39, 40].

Traditionally, non-viscous fluid dynamics is discussed in terms of the fluid velocity vector field ${\bf v}$, 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):

$$\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho{\bf v})=0,$$

(2.1)

$$\frac{\partial{\bf v}}{\partial t}+({\bf{v}}\cdot\nabla){\bf v}=-\frac{1}{\rho}\nabla p.$$

(2.2)

These are to be combined with a constitutive law giving a relationship between $p$ and $\rho$.

The above equations are already simplified versions of the general Navier-Stokes equations (which allow for viscosity terms), but we shall simplify further by working only in two dimensions and by taking the constitutive law to be the simplest possible:

$$\rho=1.$$

(2.3)

Thus $\rho$ disappears from discussion, and the equation of continuity becomes

$$\nabla\cdot{\bf v}=0.$$

(2.4)

One can get rid also of $p$, because when $\rho$ is constant then $p$ appears only as a scalar potential in Euler’s equation, this equation effectively saying that

$$\frac{\partial{\bf v}}{\partial t}+({\bf{v}}\cdot\nabla){\bf v}=\nabla({\rm something}),$$

(2.5)

where this ${\rm``something"}=p$  afterwards 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 [44], 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

$$ds^{2}=\lambda(z)^{2}|dz|^{2}=\lambda(z)^{2}(dx^{2}+dy^{2})$$

(2.6)

with $\lambda>0$. Thus the metric tensor $g_{ij}$, as appearing in general in $ds^{2}=g_{ij}dx^{i}dx^{j}$, 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)

$$\nu=\nu_{x}\,dx+\nu_{y}\,dy$$

rather than with the corresponding vector field, which then becomes

$${\bf v}=\frac{1}{\lambda^{2}}(\nu_{x}\,\frac{\partial}{\partial x}+\nu_{y}\,\frac{\partial}{\partial y}).$$

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

$$*1=\lambda^{2}(dx\wedge dy)={\rm vol}=\text{the volume (area) form},$$

$$*dx=dy,\quad*dy=-dx,\quad*(dx\wedge dy)=\lambda^{-2}.$$

Thus, for $1$-forms,

$$*\nu=-\nu_{y}\,dx+\nu_{x}\,dy,$$

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 $\mathcal{L}_{\bf v}$ of a vector field ${\bf v}$. When acting on differential forms this is related to interior derivation (or “contraction”) $i({\bf v})$ and exterior derivation $d$ by the homotopy formula

$$\mathcal{L}_{\bf{v}}=d\circ i({\bf{v}})+i({\bf{v}})\circ d.$$

(2.7)

The Hodge star and interior derivation are related by

$$i({\bf v}){\rm vol}=*\nu,$$

(2.8)

where ${\bf v}$ and $\nu$ are linked via the metric tensor as above. Thus

$$d*\nu=d(i({\bf v}){\rm vol})=\mathcal{L}_{\bf v}({\rm vol})=(\nabla\cdot{\bf v}){\rm vol}.$$

See [15] for this identity, and for further details in general. We conclude that (2.4) is equivalent to the statement that $*\nu$ is a closed form:

$$d*\nu=0.$$

(2.9)

Locally we can therefore write

$$*\nu=d\psi$$

(2.10)

for some (locally defined) stream function $\psi$. The vorticity $2$-form for a fluid is, in terms of the flow $1$-form $\nu,$

$$\omega=d\nu=(\frac{\partial\nu_{y}}{\partial x}-\frac{\partial\nu_{x}}{\partial y})\,dx\wedge dy.$$

(2.11)

The Euler equation (2.2) can be written

$$(\frac{\partial}{\partial t}+\mathcal{L}_{\bf v})({\bf\nu})=d(\frac{1}{2}|{\bf v}|^{2}-{p}).$$

(2.12)

Note that the left member involves the fluid velocity both as a vector field and as a form. The combination $\frac{\partial}{\partial t}+\mathcal{L}_{\bf v}$ can be viewed as a counterpart, for forms, to the more traditional “convective derivative” (or material derivative) $\frac{\partial}{\partial t}+{\bf{v}}\cdot\nabla$, implicitly used in (2.5), for vector fields. However, the two derivatives are not the same. The equivalence between (2.12) and (2.2) (when $\rho=1$) is a consequence of the identity

$$\mathcal{L}_{\bf v}\nu=d(\frac{1}{2}|{\bf v}|^{2})+({\bf{v}}\cdot\nabla){\nu},$$

which can be directly verified by coordinate computations. See [46, 15] for details.

Since the pressure $p$ does not appear in any other equation, the Euler equation on the form (2.12) only expresses, in analogy with (2.5), that the left member is an exact $1$-form:

$$(\frac{\partial}{\partial t}+\mathcal{L}_{\bf v})({\bf\nu})={\rm exact}=d\phi.$$

(2.13)

From the scalar $\phi$ the pressure $p$ then can be recovered via

$$\phi=\frac{1}{2}|{\bf v}|^{2}-{p}+{\rm constant}.$$

(2.14)

On acting by $d$ on (2.13), the local form of Helmholtz-Kirchhoff-Kelvin law of conservation of vorticity follows:

$$(\frac{\partial}{\partial t}+\mathcal{L}_{\bf{v}})(\omega)=0.$$

(2.15)

Since the right member of (2.13) 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

$$\frac{d}{dt}\oint_{\gamma(t)}\nu=0$$

for any closed curve $\gamma(t)$ which moves with the fluid.

In the sequel $M$ will be a closed (compact) Riemann surface provided with a Riemannian metric on the form (2.6).

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

$$(\omega_{1},\omega_{2})_{2}=\int_{M}\omega_{1}\wedge*\omega_{2}.$$

(3.1)

The given $2$-form $\omega$ then decomposes as the orthogonal decomposition of an exact form and a harmonic form:

$$\omega=d(\rm something)+{\rm harmonic}.$$

This Hodge decomposition can more precisely be written as

$$\omega=-d*dG^{\omega}+{\rm constant}\cdot{\rm vol},$$

(3.2)

where $G^{\omega}$ is normalized to be orthogonal to all harmonic $2$-forms, namely satisfying

$$\int_{M}G^{\omega}\,{\rm vol}=0.$$

(3.3)

The constant in (3.2) necessarily equals the mean value of $\omega$,

$${\rm constant}=\frac{1}{V}\int_{M}\omega,$$

(3.4)

where $V$ denotes the total volume (=area) of $M$:

$$V=\int_{M}{\rm vol}.$$

When $\omega$ is exact, as in (2.11), the second term in the right member of (3.2) disappears, hence

$$-d*dG^{\omega}=\omega$$

(3.5)

in this case. In the other extreme, when $\omega$ is harmonic, i.e. is a constant multiple of ${\rm vol}$, the first term disappears. Indeed, the whole Green function disappears:

$$G^{{\rm vol}}=0.$$

(3.6)

For $1$-forms the inner product has the same expression as for $2$-forms:

$$(\nu_{1},\nu_{2})_{1}=\int_{M}\nu_{1}\wedge*\nu_{2}.$$

(3.7)

If $\nu$ represents a fluid velocity, then $(\nu,\nu)_{1}$ has the interpretation of being (twice) the kinetic energy of the flow. For a function (potential) $u$ we consider the Dirichlet integral $(du,du)_{1}$ 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,

$$\mathcal{E}(\omega_{1},\omega_{2})=(dG^{\omega_{1}},dG^{\omega_{2}})_{1}=\int_{M}G^{\omega_{1}}\wedge\omega_{2}.$$

(3.8)

It follows, from (3.3) for example, that the volume form has no energy:

$$\mathcal{E}({\rm vol},{\rm vol})=0.$$

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

$$\int_{M}\delta_{a}\wedge\varphi=\varphi(a)$$

for every smooth function $\varphi$. Certainly $\delta_{a}$ has infinite energy, but if $a\neq b$, 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:

$$G(a,b)=G^{\delta_{a}}(b)=\mathcal{E}(\delta_{a},\delta_{b}).$$

(3.9)

Here the first equality can be taken as a definition, and then the second equality follows on using (3.2), (3.3):

	$\displaystyle\mathcal{E}(\delta_{a},\delta_{b})$	$\displaystyle=\int_{M}dG^{\delta_{a}}\wedge*dG^{\delta_{b}}=-\int_{M}G^{\delta_{a}}\wedge d*dG^{\delta_{b}}$
		$\displaystyle=\int_{M}G^{\delta_{a}}\wedge\left(\delta_{b}-\frac{1}{V}\,{\rm vol}\right)=G^{\delta_{a}}(b)=G(a,b).$

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

$$-d*dG(\cdot,w)=\delta_{w}-\frac{1}{V}{\rm{vol}}.$$

(3.10)

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 ($\mathcal{E}(\delta_{w},\delta_{w})=+\infty$, $\mathcal{E}({\rm vol},{\rm vol})=0$).

For later use we record the formulas (differentiation with respect to $z$)

$$dG(z,w)=\frac{\partial G(z,w)}{\partial z}dz+\frac{\partial G(z,w)}{\partial\bar{z}}d\bar{z}=2\operatorname{Re}\big{(}\frac{\partial G(z,w)}{\partial z}dz\big{)},\qquad$$

$$*dG(z,w)=-\mathrm{i}\frac{\partial G(z,w)}{\partial z}dz+\mathrm{i}\frac{\partial G(z,w)}{\partial\bar{z}}d\bar{z}=2\operatorname{Im}\big{(}\frac{\partial G(z,w)}{\partial z}dz\big{)}.$$

(3.13)

If $\gamma$ is any closed oriented curve in $M$ then clearly $\oint_{\gamma}dG(\cdot,w)=0$, while the conjugate period defines a function

$$U_{\gamma}(w)=\oint_{\gamma}*dG(\cdot,w)=\oint_{\gamma}\big{(}-\mathrm{i}\frac{\partial G(z,w)}{\partial z}dz+\mathrm{i}\frac{\partial G(z,w)}{\partial\bar{z}}d\bar{z}\big{)}$$

(3.14)

$$=-2\mathrm{i}\oint_{\gamma}\frac{\partial G(z,w)}{\partial z}dz=2\mathrm{i}\oint_{\gamma}\frac{\partial G(z,w)}{\partial\bar{z}}d\bar{z},$$

(3.15)

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

$$dU_{\gamma}(w)=-2\mathrm{i}\oint_{\gamma}\frac{\partial^{2}G(z,w)}{\partial{z}\partial w}d{z}\otimes dw-2\mathrm{i}\oint_{\gamma}\frac{\partial^{2}G(z,w)}{\partial{z}\partial\bar{w}}d{z}\otimes d\bar{w},$$

$$*dU_{\gamma}(w)=-2\oint_{\gamma}\frac{\partial^{2}G(z,w)}{\partial{z}\partial w}d{z}\otimes dw+2\oint_{\gamma}\frac{\partial^{2}G(z,w)}{\partial{z}\partial\bar{w}}d{z}\otimes d\bar{w}\quad$$

(integration with respect ot $z$, Hodge star and $d$ with respect to $w$). We have written out a tensor product sign to emphasize that the product between the $dz$ and $dw$ is not a wedge product. Adding and subtracting we obtain the analytic (respectively anti-analytic) differentials

$$dU_{\gamma}(w)+\mathrm{i}*dU_{\gamma}(w)=2\frac{\partial U_{\gamma}(w)}{\partial w}dw=-4\mathrm{i}\oint_{\gamma}\frac{\partial^{2}G(z,w)}{\partial{z}\partial w}d{z}\otimes dw,$$

(3.16)

$$dU_{\gamma}(w)-\mathrm{i}*dU_{\gamma}(w)=2\frac{\partial U_{\gamma}(w)}{\partial\bar{w}}d\bar{w}=-4\mathrm{i}\oint_{\gamma}\frac{\partial^{2}G(z,w)}{\partial{z}\partial\bar{w}}d{z}\otimes d\bar{w}.$$

(3.17)

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 [12] for details). Then there are corresponding harmonic differentials $dU_{\alpha_{j}}$, $dU_{\beta_{j}}$ 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 $k,j=1,\dots,\texttt{g}$,

$$\oint_{\alpha_{k}}(-dU_{\beta_{j}})=\delta_{kj},\quad\oint_{\beta_{k}}(-dU_{\beta_{j}})=0,$$

(3.18)

$$\oint_{\alpha_{k}}dU_{\alpha_{j}}=0,\qquad\oint_{\beta_{k}}dU_{\alpha_{j}}=\delta_{kj}.$$

(3.19)

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:

$$\int_{b}^{a}dU_{\alpha_{j}}=\oint_{\alpha_{j}}*dG^{\delta_{a}-\delta_{b}},$$

(3.20)

$$\int_{b}^{a}dU_{\beta_{j}}=\oint_{\beta_{j}}*dG^{\delta_{a}-\delta_{b}}.$$

(3.21)

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 $U_{\gamma}$.

When $\nu$ is the flow $1$-form of an incompressible fluid, the equation of continuity says that $d*\nu=0$ (see (2.9)). The vorticity $2$-form is $\omega=d\nu$, and (3.5) holds. Thus $d(\nu+*dG^{\omega})=0$, and setting

$$\eta=\nu+*dG^{\omega}$$

(4.1)

it follows that $\eta$ is harmonic: $d\eta=0=d*\eta$. Locally we can write $*\eta=d\psi_{0}$ for some harmonic function $\psi_{0}$. Then

$$\psi=G^{\omega}+\psi_{0}$$

(4.2)

becomes a locally defined stream function for the flow, so that

$$*\nu=d\psi=dG^{\omega}+d\psi_{0}.$$

Different local choices of $\psi$ may differ by additive time dependent constants. The roles of the additional terms $\eta$ and $\psi_{0}$, complementing the contribution from the Green function, will be somewhat clarified in the context of planar vortex motion and the hydrodynamic Green function in Section 9.

The relation (4.1) written on the form $\nu=\eta-*dG^{\omega}$ is an orthogonal decomposition of the flow $1$-form $\nu$ with respect to the inner product (3.7). Indeed,

$$(\eta,-*dG^{\omega})_{1}=\int_{M}\eta\wedge dG^{\omega}=-\int_{M}d\eta\wedge G^{\omega}=0,$$

since $\eta$ is harmonic. It follows that (twice) the total (kinetic) energy $(\nu,\nu)_{1}$ of the flow is given by

$$2(\nu,\nu)_{1}=\int_{M}\nu\wedge*\nu=\frac{\mathrm{i}}{2}\int_{M}(\nu+\mathrm{i}*\nu)\wedge(\nu-\mathrm{i}*\nu)=$$

$$=\int_{M}dG^{\omega}\wedge*dG^{\omega}+\int_{M}\eta\wedge*\eta=\mathcal{E}(\omega,\omega)+\int_{M}\eta\wedge*\eta=$$

$$=\int_{M}G^{\omega}\wedge\omega+\sum_{j=1}^{\texttt{g}}(\oint_{\alpha_{j}}\eta\oint_{\beta_{j}}*\eta-\oint_{\alpha_{j}}*\eta\oint_{\beta_{j}}\eta).$$

We shall now specialize on the point vortex case, with 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

$$\Gamma=\sum_{k=1}^{n}\Gamma_{k}$$

(4.3)

may be non-zero and there will then be a compensating uniform counter vorticity. The total vorticity $\omega=d\nu$, which satisfies $\int_{M}\omega=0$, appears as the right member in

$$-d*dG^{\omega}=\omega=\sum_{k=1}^{n}\Gamma_{k}\delta_{w_{k}}-\frac{\Gamma}{V}{\rm vol}.$$

(4.4)

Explicitly we have, on recalling (3.6),

$$G^{\omega}(z)=\sum_{k=1}^{n}\Gamma_{k}{G(z,w_{k})}.$$

(4.5)

On using (3.13) the conjugate $\alpha$-periods of $G^{\omega}$ can be expressed as

$$\oint_{\alpha_{j}}*d{G^{\omega}}=-\sum_{k=1}^{n}2\mathrm{i}\Gamma_{k}\oint_{\alpha_{j}}\frac{\partial G(z,w_{k})}{\partial z}dz.$$

(4.6)

By differentiation and using also (3.15), (3.16) we have

$$\frac{\partial}{\partial w_{k}}\oint_{\alpha_{j}}*d{G^{\omega}}=\Gamma_{k}\frac{\partial U_{\alpha_{j}}(w_{k})}{\partial w_{k}}=\mathrm{i}\Gamma_{k}\frac{\partial U^{*}_{\alpha_{j}}(w_{k})}{\partial w_{k}},$$

(4.7)

$$\frac{\partial}{\partial w_{k}}\big{(}\oint_{\alpha_{j}}*d{G^{\omega}}\big{)}dw_{k}=\frac{\Gamma_{k}}{2}(dU_{\alpha_{j}}(w_{k})+\mathrm{i}*dU_{\alpha_{j}}(w_{k})).$$

(4.8)

Here $U^{*}_{\alpha_{j}}$ denotes a harmonic conjugate of $U_{\alpha_{j}}$, whereby $*dU_{\alpha_{j}}=d(U^{*}_{\alpha_{j}})$. 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,

$$2(\nu,\nu)_{1}=\sum_{k,j=1}^{n}\Gamma_{k}\Gamma_{j}G(w_{k},w_{j})+\sum_{j=1}^{\texttt{g}}\big{(}\oint_{\alpha_{j}}\eta\oint_{\beta_{j}}*\eta-\oint_{\alpha_{j}}*\eta\oint_{\beta_{j}}\eta\big{)}.$$

However, the presence of the terms $\Gamma_{k}^{2}G(w_{k},w_{k})$ makes the first term become infinite. In order to renormalize this singular behavior we isolate the logarithmic pole in the Green function by writing

$$G(z,w)=\frac{1}{2\pi}(-\log|z-w|+H(z,w)),$$

(4.9)

and expand, for a fixed $w$, the regular part in a power series in $z$ as

$$H(z,w)=h_{0}(w)+\frac{1}{2}\left(h_{1}(w)(z-w)+\overline{h_{1}(w)}(\bar{z}-\bar{w})\right)+$$

(4.10)

$$+\frac{1}{2}\left(h_{2}(w)(z-w)^{2}+\overline{h_{2}(w)}(\bar{z}-\bar{w})^{2}\right)+h_{11}(w)(z-w)(\bar{z}-\bar{w})+\mathcal{O}(|z-w|^{3}).$$

In Appendix 2, Section 11, it is discussed how the coefficients in the expansion (4.10) behave under conformal mapping. For example, the coefficient $h_{11}(w)$ transforms as the density of a metric, and it is indeed proportional to the given metric:

$$h_{11}(w)=\frac{\pi}{2V}\lambda(w)^{2}.$$

(4.11)

See (11.5). The coefficients $h_{0}(w)$, $h_{1}(w)$ and $h_{2}(w)$ transform, in certain combinations, as “connections” under conformal mappings. For $h_{0}(w)$ this amounts to saying that it too defines a metric, namely the Robin metric, via

$$ds=e^{-h_{0}(w)}|dw|.$$

(4.12)

Metrics of this kind are implicit in the theory of capacity functions, as exposed in [42]. 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 various given 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 need be replaced by a term which counteracts the inhomogeneous transformation law of $h_{0}(w)$ (see (11.8)). Such a term comes naturally from the given metric $ds=\lambda(w)|dw|=e^{\log\lambda(w)}|dw|$. We see that minus $\log\lambda(w)$ has the right properties, and it combines with $h_{0}(w)$ into

$$R_{\rm robin}(w)=\frac{1}{2\pi}(h_{0}(w)+\log\lambda(w)).$$

(4.13)

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:

$$G(z,w)=-\frac{1}{2\pi}\log{\rm dist}(z,w)+R_{\rm robin}(w)+\mathcal{O}({\rm dist}(z,w)).$$

As for the infinite kinetic energy, we conclude that it should be renormalized by replacing the diagonal terms $G(w_{k},w_{k})$ by $R_{\rm robin}(w_{k})$. This gives the same equations of motion as more direct approaches, or those available in the literature, like [26, 3, 10]. Thus (twice) the renormalized energy is

$$2(\nu,\nu)_{1,{\rm renorm}}=\sum_{k=1}^{n}\Gamma_{k}^{2}R_{\rm robin}(w_{k})+\sum_{k\neq j}\Gamma_{k}\Gamma_{j}G(w_{k},w_{j})+$$

$$+\sum_{j=1}^{\texttt{g}}\big{(}\oint_{\alpha_{j}}\eta\oint_{\beta_{j}}*\eta-\oint_{\alpha_{j}}*\eta\oint_{\beta_{j}}\eta\big{)}.$$

(4.14)

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 (4.13), 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 (4.14), 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 (Sections 7 and 8). Then only the local terms survive.

We need to elaborate further the expression (4.1) and relate it to the global circulations. The $1$-form $\eta$ is harmonic and hence can be expanded as

$$\eta=-\sum_{j=1}^{\texttt{g}}A_{j}dU_{\beta_{j}}+\sum_{j=1}^{\texttt{g}}B_{j}dU_{\alpha_{j}}.$$

(4.15)

The coefficients $A_{k}$, $B_{k}$ are in view of (3.18), (3.19) given by

$$A_{k}=\oint_{\alpha_{k}}\eta,\quad B_{k}=\oint_{\beta_{k}}\eta.$$

The circulations of the total flow $\nu=\eta-*dG^{\omega}$ then become

$$a_{k}=\oint_{\alpha_{k}}\nu=A_{k}-\oint_{\alpha_{k}}*dG^{\omega},$$

$$b_{k}=\oint_{\beta_{k}}\nu=B_{k}-\oint_{\beta_{k}}*dG^{\omega}.$$

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 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 (2.13), and in the notation used there, the circulation of $\nu$ around $\gamma$ changes with speed

$$\frac{d}{dt}\oint_{\gamma}\nu=\oint_{\gamma}\frac{\partial\nu}{\partial t}=\oint_{\gamma}(d\phi-\mathcal{L}_{\bf v}\nu)=-\oint_{\gamma}i({\bf v})d\nu=$$

$$=-\oint_{\gamma}i({\bf v})\omega=\oint_{\gamma}i({\bf v})\big{(}\frac{\Gamma}{V}{\rm vol}\big{)}=\frac{\Gamma}{V}\oint_{\gamma}*\nu.$$

Here we used also (2.7), (2.8), (4.4). It follows in particular, on choosing $\gamma=\alpha_{k},\beta_{k}$, that

$$\frac{da_{k}}{dt}=\frac{\Gamma}{V}\oint_{\alpha_{k}}*\nu=\frac{\Gamma}{V}\oint_{\alpha_{k}}*\eta,$$

(4.16)

$$\frac{db_{k}}{dt}=\frac{\Gamma}{V}\oint_{\beta_{k}}*\nu=\frac{\Gamma}{V}\oint_{\beta_{k}}*\eta.$$

(4.17)

The period matrix (written here in block form)

$$\left(\begin{array}[]{cc}P&R\\ R^{T}&Q\\ \end{array}\right)=\left(\begin{array}[]{cc}(-\oint_{\beta_{k}}*dU_{\beta_{j}})&(\oint_{\beta_{k}}*dU_{\alpha_{j}})\\ (\oint_{\alpha_{k}}*dU_{\beta_{j}})&(-\oint_{\alpha_{k}}*dU_{\alpha_{j}})\\ \end{array}\right)=$$

$$=\left(\begin{array}[]{cc}(\int_{M}dU_{\beta_{k}}\wedge*dU_{\beta_{j}})&(-\int_{M}dU_{\beta_{k}}\wedge*dU_{\alpha_{j}})\\ (-\int_{M}dU_{\alpha_{k}}\wedge*dU_{\beta_{j}})&(\int_{M}dU_{\alpha_{k}}\wedge*dU_{\alpha_{j}})\\ \end{array}\right)$$

(4.18)

is symmetric and positive definite (see [12] 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, $j$ the column index. Thus, for example, $R_{kj}=\oint_{\beta_{k}}*dU_{\alpha_{j}}$.

Next we write

$$dU_{\alpha}=\left(\begin{array}[]{c}dU_{\alpha_{1}}\\ \vdots\\ dU_{\alpha_{\texttt{g}}}\end{array}\right),\quad dU_{\beta}=\left(\begin{array}[]{c}dU_{\beta_{1}}\\ \vdots\\ dU_{\beta_{\texttt{g}}}\end{array}\right).$$

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 $*dU_{\alpha}$, $*dU_{\beta}$, in similar matrix notation. The relation between the bases is

We arrange also the circulations of the flow $\nu$ into column vectors:

$$a=\left(\begin{array}[]{c}a_{1}\\ \vdots\\ a_{\texttt{g}}\end{array}\right)=\left(\begin{array}[]{c}\oint_{\alpha_{1}}\nu\\ \vdots\\ \oint_{\alpha_{\texttt{g}}}\nu\end{array}\right),\quad b=\left(\begin{array}[]{c}b_{1}\\ \vdots\\ b_{\texttt{g}}\end{array}\right)=\left(\begin{array}[]{c}\oint_{\beta_{1}}\nu\\ \vdots\\ \oint_{\beta_{\texttt{g}}}\nu\end{array}\right),$$

briefly written as

$$a=\oint_{\alpha}\nu,\quad b=\oint_{\beta}\nu.$$

(4.20)

Similarly for other vectors of circulations, for example

$$A=\oint_{\alpha}\eta=\oint_{\alpha}\nu+\oint_{\alpha}*dG^{\omega}=a+\oint_{\alpha}*dG^{\omega},$$

(4.21)

$$B=\oint_{\beta}\eta=\oint_{\beta}\nu+\oint_{\beta}*dG^{\omega}=b+\oint_{\beta}*dG^{\omega}.$$

(4.22)

The $\alpha$-periods of the conjugated Green function were computed in (4.6), and from (4.7), (4.8) it follows that

$$\frac{\partial A}{\partial w_{k}}dw_{k}+\frac{\partial A}{\partial\bar{w}_{k}}d\bar{w}_{k}={\Gamma_{k}}\,dU_{\alpha}(w_{k}).$$

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

$$dA=da+d\oint_{\alpha}*dG^{\omega}=da+\sum_{k=1}^{n}\Gamma_{k}dU_{\alpha}(w_{k}),$$

(4.23)

$$dB=db+d\oint_{\beta}*dG^{\omega}=db+\sum_{k=1}^{n}\Gamma_{k}dU_{\beta}(w_{k}).$$

(4.24)