On the Interplay Between Vortices and Harmonic Flows: Hodge Decomposition of Euler’s Equations in 2d

Abstract

Let $\Sigma$ be a compact manifold without boundary whose first homology is nontrivial. The Hodge decomposition of the incompressible Euler equation in terms of 1-forms yields a coupled PDE-ODE system. The $L^{2}$-orthogonal components are a “pure” vorticity flow and a potential flow (harmonic, with the dimension of the homology). In this paper we focus on $N$ point vortices on a compact Riemann surface without boundary of genus $g$, with a metric chosen in the conformal class. The phase space has finite dimension $2N+2g$. We compute a surface of section for the motion of a single vortex ($N=1$) on a torus ($g=1$) with a nonflat metric that shows typical features of nonintegrable 2 degrees of freedom Hamiltonians. In contradistinction, for flat tori the harmonic part is constant. Next, we turn to hyperbolic surfaces ($g\geqslant 2$) having constant curvature $-1$, with discrete symmetries. Fixed points of involutions yield vortex crystals in the Poincaré disk. Finally, we consider multiply connected planar domains. The image method due to Green and Thomson is viewed in the Schottky double. The Kirchhoff – Routh Hamiltonian given in C. C. Lin’s celebrated theorem is recovered by Marsden – Weinstein reduction from $2N+2g$ to $2N$. The relation between the electrostatic Green function and the hydrodynamic Green function is clarified. A number of questions are suggested.

APPENDIX A. HODGE DECOMPOSITION

Hodge Theory for Compact Manifolds $\Sigma^{n}$ Without Boundary

See [A1] for a complete exposition. A metric $g$ in $T\Sigma$ extends to tensors and forms via Gramians:

$$\langle\nu_{2},\nu_{2}\rangle_{Hodge}:=\int_{\Sigma}\nu_{1}\wedge\star\nu_{2},$$

(A.1)

where the operator $\star\nu$ from $k$ to $(n-k)$ forms such that $(\star)^{2}=(-1)^{k(n-k)}$ can be defined by

$$\xi\wedge\star\nu=g(\xi,\nu)\mu \quad(\mu=\text{volume form}).$$

(A.2)

Theorem (Hodge).

There is an orthogonal decomposition of any k-form $\nu\in\Omega^{k}(\Sigma)$ as a sum

$$\nu=d\alpha\oplus\delta\Psi\oplus\eta,$$

$$\alpha\in\Omega^{k-1}(\Sigma),\quad\Psi\in\Omega^{k+1}(\Sigma),\quad\eta\in\text{ kernel}(\Delta),$$

• i) $\delta:\Omega^{k+1}(\Sigma)\to\Omega^{k}(\Sigma)$, given by $\delta=(-1)^{nk+1}\star d\star$ is the adjoint of $d$: $\langle d\nu_{1},\nu_{2}\rangle=\langle\nu_{1},\delta\nu_{2}\rangle$.

• ii) $\Delta=d\delta+\delta d$ is an elliptic operator.

  $$\text{ kernel}(\Delta)={\rm Harm}^{k}(\Sigma)\textit{ is of finite dimension, and determines the cohomology}.$$

• iii) $\Delta\eta=0$ iff both $d\eta=0$ and $\delta\eta=0$ and $\left\{d\alpha\oplus\delta\omega\right\}={\rm Image}(\Delta)(\perp\quad{\rm to}\quad{\rm kernel}(\Delta)).$

Schematically,

$$\Omega^{k}(\Sigma)={\rm im}d\oplus{\rm ker}\delta={\rm im}\delta\oplus{\rm ker}d={\rm im}d\oplus{\rm im}\delta\oplus{\rm ker\Delta}={\rm im}\Delta\oplus{\rm ker}\Delta.$$

Manifolds with Boundary: Hodge – Morrey – Friedrichs [A2, A3, A4]. For Computational Implementation, See [A5, A6]. For Application in Blood Physiology, See [A7, A8, A9]

We have not explored this situation in this paper. In order to show why the Hodge theorem has to be refined, observe that, if $\Sigma$ has boundary, then it is possible for a 1-form to be in the kernel of the Laplacian without being both closed and co-closed.

A simple example: Let $\Sigma=A$ be the annulus $A:0<a\leqslant r\leqslant b$. The 2-form $\omega=(1/2)\log(x^{2}+y^{2})dx\wedge dy$ is harmonic (since $\log r$ is harmonic in the annulus). It is actually also exact: $\omega=d[\int^{r}r\log r^{2} d\theta].$ Notice that it is not co-closed: $\phi:=\delta\omega=(-ydx+xdy)/(x^{2}+y^{2})$represents the 1-dimensional cohomology ${\rm Harm}^{1}(A)$.

Hodge Theory in 2d: Formalism

For brevity we will omit the wedge $\wedge$ and sometimes denote by $\mu$ the area form. The total area will be denoted by $V$. All coordinate systems $(x,y)$ will be isothermal, with $ds^{2}=\lambda^{2}(dx^{2}+dy^{2}).$

We will circumvent the complex notation $dz$ and $d\bar{z}$, and write

$$\nu=\nu_{x}dx+\nu_{y}dy,\quad v=\nu^{\sharp}=(1/\lambda^{2})(\nu_{x}\partial_{x}+\nu_{y}\partial_{y}).$$

(A.3)

The main rules of the game are $\star\nu=-v_{y}dx+\nu_{x}dy$ (90 degrees rotation), together with

$$\begin{gathered}\displaystyle\star 1=\lambda^{2}dx\wedge dy=\mu,\quad\star dx=dy,\quad\star dy=-dx,\quad\star\mu=1,\\ \displaystyle\star\star\nu=-\nu\ \text{(for 1-forms)},\quad\delta=-\star d\star\ \text{(any degree)},\quad\Delta=d\delta+\delta d\ \text{commutes with $d$ and $\delta$},\\ \displaystyle\Delta f=-\frac{1}{\lambda^{2}}(f_{xx}+f_{yy})\quad\text{(for functions)}.\end{gathered}$$

The following formulas are useful in computations with 1-forms:

$$\star\nu=i_{v}\mu,L_{v}=di_{v}+i_{v}d\Rightarrow d(\star\nu)=d(i_{v}\mu)=L_{v}\mu=:({\rm div}v)\mu.$$

(A.4)

All our vector fields will be incompressible: $d\star\nu=0$. This assumption is used to say that locally

$$\star\nu=d\psi.$$

(A.5)

Proposition.

The inner product of $1$-forms requires just the complex structure.

Proof.

By definition, $\langle\nu_{1},\nu_{2}\rangle=\int_{\Sigma}\nu_{1}\wedge\star\nu_{2}.$ Locally, $\nu_{1}=a_{1}dx+a_{2}dy,\nu_{2}=b_{1}dx+b_{2}dy$, then

$$\langle\nu_{1},\nu_{2}\rangle=\int(a_{1}b_{1}+a_{2}b_{2})dxdy.$$

If one goes back to the vector fields, $\nu_{1}^{\sharp}=(a_{1}\partial_{x}+a_{2}\partial_{y})/\lambda^{2},\nu_{2}^{\sharp}=(b_{1}\partial_{x}+b_{2}\partial_{y})/\lambda^{2}$ and

$$\langle\nu_{1},\nu_{2}\rangle=(a_{1}b_{1}+a_{2}b_{2})\lambda^{2}/\lambda^{4},$$

because $|\partial_{x}|=\lambda$, etc. So

$$ \langle \nu_1^\sharp, \nu_2^\sharp \rangle = \int \frac{a_1 b_1 + a_2 b_2}{{\hspace{3mm}\big{/}}{\hspace{-3mm}\lambda^4}} {\hspace{2mm}\big{/}}{\hspace{-3mm}\lambda^2} {\hspace{2mm}\big{/}}{\hspace{-5mm}(\lambda^2} dx dy). $$

Thus, there is no need to refer to a specific metric for forms, although for the inner product between vector fields one has to specify the metric tensor. As we have seen, things miraculously compensate:

Proposition.

The harmonicity of $1$-forms, namely, the two conditions $d\nu=0,d\star\nu=0$, does not depend on the chosen metric in the conformal class.

APPENDIX B. BIOT – SAVART

Two Dimensions

Recall that $\Psi$ is the Hodge star of the Green function $G^{\omega}$ (notations in Section 2.1) and the Green functions can in principle be constructed by potential theoretic methods (minimization of energy, Perron’s method of subharmonic functions, etc).

For 2-forms in dimension two one has $\Delta = d \delta + {\hspace{2mm}\big{/}}{\hspace{-4mm}\delta d } = d \delta$, since the second term disappears by default: $d$ is applied in the maximal dimension of the manifold.

Given a vorticity 2-form $\omega$, write Poisson’s equation (for functions), $\Delta\psi=\star\omega$ where $\psi$ is a function. By abuse of language one can omit the $\star$ in front of the 2-form $\omega$ (so $\omega$ is interpreted as a density multiplying $\mu$, just to conform with the fluid mechanics notation).

The requirement for the solution $\psi$ to exist is $\omega$ to be exact (it is a vorticity 2-form)

$$\Bigl{(}\omega\text{ is exact }\Leftrightarrow\omega\text{ is }\perp\text{ to harmonic 2-forms (constant multiples of $\mu$)}\Leftrightarrow\int_{\Sigma}\omega=0\Bigr{)}.$$

Let $\Psi=\star\psi=\psi\mu.$ Consider now $\delta\Psi$ (which is purely vortical by definition). Then $d (\delta \Psi) = \Delta \Psi - \delta {\hspace{2mm}\big{/}}{\hspace{-4mm} d \Psi} = \omega .$

Note that the vortical part of the energy can be rewritten as

$$H_{vort}(\omega):=H(\delta\Psi)=\frac{1}{2}\langle\delta\Psi,\delta\Psi\rangle=\frac{1}{2}\langle d\delta\Psi,\Psi\rangle=\frac{1}{2}\langle\Delta\Psi,\Psi\rangle=\frac{1}{2}\langle\omega,\Delta^{-1}\omega\rangle.$$

Note that $\Psi$ is not unique. One can add any $\Psi_{o}$ such that $\delta\Psi_{o}=0$. This means that $\Psi_{o}$ is a constant multiple of $\mu$. One may normalize $\Psi=\Psi_{\omega}$ by requiring that it be orthogonal to all harmonic functions (namely, the constants), i. e., by requiring that

$$\int_{\Sigma}\Psi_{\omega}=0.$$

Then, by identification,

$$\nu=\delta\Psi_{\omega}+\eta=-\star d\star\Psi_{\omega}+\eta=-\star dG^{\omega}+\eta$$

so the Hodge star of $\Psi_{\omega}$ is the Green function: $G^{\omega}=\star\Psi_{\omega}$ with normalization

$$\int_{\Sigma}G^{\omega}\mu=0.$$

Extending the Hodge decomposition to certain singular forms gives, for example,

$$G(s,r)=G^{\delta_{r}\mu}(s)=\star\Psi_{\delta_{r}\mu}(s).$$

Biot – Savart in ${\mathbb{R}}^{n},n\geqslant 3$

In dimension greater than two, for 2-forms, the term $\delta d$ in $\Delta=d\delta+\delta d$ no longer vanishes, so we cannot use Poisson’s equation anymore. The Biot – Savart law in ${\mathbb{R}}^{3}$ is taught in basic electromagnetism classes. For ${\mathbb{R}}^{4}$, see [B3]. For ${\mathbb{R}}^{n},n\geqslant 4$ we just found a very recent reference [B1]. One of the authors (BG) formulated a version of his own in [B2], Lemma 1.3.

Let $f=(f_{1},\dots,f_{n})$ be a vector field in ${\mathbb{R}}^{n}$, vanishing sufficiently fast at infinity (in order for the convolutions to make sense). No regularity of $f$ is assumed, we rather think of a distributional setting, allowing situations in which $f$ is singular, such as having support on lower-dimensional manifolds (lines, surfaces, etc). Let

$$E(x)=\frac{c_{n}}{|x|^{n-2}}\quad(n\geqslant 3)$$

(B.1)

be the standard fundamental solution of the Laplacian, $c_{N}$ chosen so that $\Delta E=\delta_{o}$.

Proposition 3.

With $\star=\star_{\rm conv}$ denoting convolution, we have the Biot – Savart type representation:

$$f=-({\rm div}f)\star\nabla E-({\rm curl}f)\star\nabla E.$$

(B.2)

The componentwise interpretation of this will be clear from the derivation below, where $E_{j}=\partial E/\partial x_{j}$ $($the components of $\nabla E)$. For each $k=1,\dots,n$ we have, using that the derivative of a convolution can be moved to an arbitrary factor in it, and also that $\partial E_{j}/\partial x_{k}=\partial E_{k}/\partial x_{j}$:

$$ \begin{aligned} f_k & =-(\Delta f_k)*E= -\sum_{j=1}^n \frac{\partial^2 f_k}{\partial x_j^2}\star E= -\sum_{j=1}^n \frac{\partial f_k}{\partial x_j}\star E_j \\ & = -\sum_{j=1}^n \frac{\partial}{\partial x_j}(f_k\star E_j)+\sum_{j=1}^n \frac{\partial}{\partial x_k}(f_j\star E_j)-\sum_{j=1}^n \frac{\partial}{\partial x_j}(f_j\star E_k)\\ & =-\sum_{j=1}^n \frac{\partial f_j}{\partial x_j}\star E_k-\sum_{j=1}^n \Big(\frac{\partial f_k}{\partial x_j}-\frac{\partial f_j}{\partial x_k}\Big)\star E_j. \end{aligned} $$

(B.3)

The ${\rm curl}$ of the vector field $f$ is implicit in the second term of this formula. It is an antisymmetric tensor corresponding to the exterior differential acting on a one-form.

Case $n=3$. The above reduces to the ordinary Biot – Savart law. We have $f=\left(f_{1},f_{2},f_{3}\right)$:

$$ \begin{aligned} {\rm curl} f = & \left( \frac{\partial f}{\partial x_2} - \frac{\partial f}{\partial x_3} , \frac{\partial f}{\partial x_3} - \frac{\partial f}{\partial x_1} , \frac{\partial f}{\partial x_1} - \frac{\partial f}{\partial x_2} \right) \\ {\rm div} f = & \frac{\partial f}{\partial x_1} + \frac{\partial f}{\partial x_2} + \frac{\partial f}{\partial x_3}, \nabla E = (E_1,E_2,E_3) = - \frac{1}{4\pi} \frac{x}{|x|^3} \end{aligned} $$

as vector and scalar fields in ${\mathbb{R}}^{3}$. Spelling out formula (B.3) in this vector analysis language results in

$$f(x)=\frac{1}{4\pi}\int_{{\mathbb{R}}^{3}}({\rm div}f)(y)\frac{x-y}{|x-y|^{3}}d^{3}y+\frac{1}{4\pi}\int_{{\mathbb{R}}^{3}}({\rm curl}f)(y)\times\frac{x-y}{|x-y|^{3}}d^{3}y,$$

(B.4)

where $d^{3}y=dy_{1}dy_{2}dy_{3}$ and $\times$ denotes the ordinary vector product. When $f$ is a stationary magnetic field, so that ${\rm div}f=0$ and ${\rm curl}f=$ current density, by Maxwell’s equations, the above formula agrees with Biot – Savart’s law found in textbooks. If $f$ instead represents a stationary electric field, then ${\rm curl}f=0$ and we obtain Coulomb’s law for an extended charge distribution ${\rm div}f.$

Formulation for One-Forms

Next, we wish to reformulate (B.3) in a way which in principle opens up for generalizations to curved manifolds. We then consider $f$ as a differential form rather than a vector field. Thus,

$$f=f_{1}dx_{1}+\cdots+f_{n}dx_{n}.$$

Proposition.

We have

$$f=dE\star_{\rm conv}\delta f-i_{(dE)^{\sharp}}\star_{\rm conv}df.$$

(B.5)

The second term shall be interpreted as “interior convolution product”.

Proof.

Noting that, in our Cartesian context,

$$ \begin{aligned} df & = \sum_{k,j=1}^n \frac{\partial f_k}{\partial x_j} dx_j \wedge dx_k = \frac{1}{2} \left(\frac{\partial f_k}{\partial x_j} - \frac{\partial f_j}{\partial x_k} \right) dx_j \wedge dx_k \\ \delta f & = - d \star d \star f = - \sum_{k=1}^n \frac{\partial f_k}{\partial x_k } = - {\rm div} f \\ & \hspace{-3mm} i_{(dE)^{\sharp}} (dx_j \wedge dx_k ) = E_j dx_k - E_k dx_j, \end{aligned} $$

(B.6)

we have by (B.3):

$$ \begin{aligned} F &= - \delta f \star_{\rm conv} dE - \sum_{k,j=1}^n \left(\frac{\partial f_k}{\partial x_j} - \frac{\partial f_j}{\partial x_k} \right) \star_{\rm conv} E_j dx_k \\ &= \delta f \star_{\rm conv} dE - \frac{1}{2} \left(\frac{\partial f_k}{\partial x_j} - \frac{\partial f_j}{\partial x_k} \right) \star_{\rm conv} \left( E_j dx_k - E_k dx_j \right) \\ &= \delta f \star_{\rm conv} dE - \frac{1}{2} \left(\frac{\partial f_k}{\partial x_j} - \frac{\partial f_j}{\partial x_k} \right) \left( i_{(dE)^{\sharp}} \star_{\rm conv} dx_j \wedge dx_k \right) \\ &= dE \star_{\rm conv} \delta f - i_{(dE)^{\sharp}} \star_{\rm conv} df. \end{aligned} $$

(B.7)

For curved manifolds one has to replace the convolutions above, which can be interpreted as different kinds of Newtonian potentials, by expressions involving Green potentials. For example, with a $0$-form $\rho,E\star_{\rm conv}\rho$ would be written $E^{\rho}$, or $G^{\rho}$ with $G$ some Green function.

APPENDIX C. DIRECT PROOF OF LEMMA 3

We first show that the function

$$U_{\gamma}(s)=\oint_{\gamma}\star_{r}d_{r}G(r,s)\quad(\text{integral in }r)$$

(C.1)

is harmonic if $s\not\in\gamma$. The Laplace operator with respect to $s$ can be written as

$$\Delta_{s}G(r,s)=-\star_{s}d_{s}\star_{s}d_{s}G(r,s)\quad(\text{see Appendices A and B).}$$

The use of local isothermal coordinates $s=(x,y)$ implies:

$$\mu_{s}=\lambda^{2}(x,y)dx\wedge dy,\quad\star\mu_{s}=1,\quad\Delta_{s}G(r,s)=-\lambda^{2}(x,y)\bigl{(}\partial_{x}^{2}+\partial_{y}^{2}\bigr{)}G(r,x,y).$$

Clearly,

$$\Delta_{s}\star_{r}d_{r}G(r,s)=\star_{r}d_{r}\Delta_{s}G(r,s){\rm if}r\neq s.$$

The symmetry of the Green function $G(r,s)=G(s,r)$ and Eq. (3.10) implies that

$$\Delta_{s}G(r,s)=-V^{-1}{\rm for}r\neq s.$$

Therefore, if $s\not\in\gamma$, then

$$\Delta_{s}U_{\gamma}(s)=\oint_{\gamma}\Delta_{s}\star_{r}d_{r}G(r,s)=\oint_{\gamma}\star_{r}d_{r}\Delta_{s}G(r,s)=-\oint_{\gamma}\star_{r}d_{r}(1/V)=0.$$

(C.2)

We now study $U_{\gamma}(s)$ in a neighborhood of a point in $\gamma$. We assume that $\gamma$ is a real analytic curve. In this case, any $s_{0}\in\gamma$ has a neighborhood $N$ in which there are isothermal coordinates $(x,y):N\to{\mathbb{C}}$ such that

$$\gamma\cap N=\{|x|<a,y=0\}{\rm and}(x,y)(s_{0})=(0,0).$$

The orientation of $\gamma$ coincides with the orientation of the $x$-axis. Let $s$ be a point in $N$ with coordinates $(x,y)(s)=(\xi,\eta)$ such that

$$U_{\gamma}(s)=\int_{\gamma\cap N}\star_{r}d_{r}G(r,s)+\int_{\gamma-\gamma\cap N}\star_{r}d_{r}G(r,s).$$

Since $G(r,s)$ is analytic for $r\in\{\gamma-\gamma\cap N\}$ and $s\in N$, the second integral is an analytic function of $s$. Therefore, the singular behavior of $U_{\gamma}(s)$ is determined by

$$\int_{\gamma\cap N}\star_{r}d_{r}G(r,s)=\int_{-a}^{a}-\partial_{y}G(r,s)dx.$$

In the isothermal coordinates

$$G(r,s)=-\frac{1}{4\pi}\log\big{(}(x-\xi)^{2}+(y-\eta)^{2}\big{)}+h(x,y,\xi,\eta),$$

where $h$ is an analytic function on $N$. The singular behavior of $U_{\gamma}(s)$ is, therefore, determined by the limit

$$ \begin{aligned} & \frac{1}{4\pi} \int_{-a}^a \partial_y \log\big((x-\xi)^2+(y-\eta)^2\big) \Big|_{y=0}dx \\ {}& \quad = - \frac{1}{2\pi} \int_{-a}^a\frac{\eta }{(x-\xi)^2+\eta^2}dx\\ {}& \quad = -\frac{1}{2\pi}\bigg\{ \arctan\left(\frac{a-\xi}{\eta}\right)+ \arctan \left(\frac{a+\xi}{\eta}\right)\bigg\}. \end{aligned} $$

We conclude that for $s\in N$

$$U_{\gamma}(\xi,\eta)=-\frac{1}{2\pi}\bigg{\{}\arctan\left(\frac{a-\xi}{\eta}\right)+\arctan\left(\frac{a+\xi}{\eta}\right)\bigg{\}}+U_{R}(\xi,\eta),$$

(C.3)

where $U_{R}$ is an analytic function.

Let $(\xi,0)$ be a point in $\gamma\cap N$, which implies $|\xi|<a$, and make $(\xi,\eta)\to(\xi,0_{-})$ from the right-hand side of $\gamma$, namely, with $\eta<0$, then

$$U(\xi,0_{-})=\lim_{\eta\to 0_{-}}U_{\gamma}(\xi,\eta)=\frac{1}{2}+U_{R}(\xi,0).$$

(C.4)

If the limit is taken from the left-hand side of $\gamma$, namely, with $\eta>0$, then

$$U(\xi,0_{+})=\lim_{\eta\to 0_{+}}U_{\gamma}(\xi,\eta)=-\frac{1}{2}+U_{R}(\xi,0).$$

(C.5)

Therefore $U(\xi,0_{-})-U(\xi,0_{+})=1$ and the function $U_{\gamma}(s)$ jumps by one when $\gamma$ is crossed from the left to the right. The differential of $U_{\gamma}$ as given in Eq. (C.3) is $dU_{R}$ plus

$$-2a\frac{2\xi\eta d\xi+\left(a^{2}-\xi^{2}+\eta^{2}\right)d\eta}{\left((a-\xi)^{2}+\eta^{2}\right)\left((a+\xi)^{2}+\eta^{2}\right)},$$

which is analytic over $\gamma$. Therefore, $dU_{\gamma}$ is a harmonic differential that is regular over $\gamma$.     $\square$

This was presented just as a guide for the nonexpert. In different degrees of sophistication such an analysis can be found, for example, in Hermann Weyl’s classical 1913 treatise [C1], and it is also similar to jump formulas for Cauchy integrals [C2, C3]. The Cauchy integral of an analytic $f$ in a domain $D$ equals $f$ inside the domain and zero outside, hence the jump across the boundary is exactly $f$. With $f=1$ we are integrating just the Cauchy kernel, and the singularity of this is of the same type as that of $dG$ (or rather $dG+i\star dG$). In complex analysis, for the Cauchy integral, the jump formula often goes under the name Sokhotski – Plemelj jump formula (tracing back to 1868).