Research ContributionArnold Mathematical Journal

Received: 3 February 2016 / Revised: 21 August 2016 / Accepted: 15 October 2016

# The Geometry of Axisymmetric Ideal Fluid Flows with Swirl

Boulder CO USA,
Stephen C. PrestonBrooklyn College,
Brooklyn NY USA,
New York NY USA,
Stephen.Preston@brooklyn.cuny.edu
The second author gratefully acknowledges support from NSF Grants DMS-1157293 and DMS-1105660

### Abstract

The sectional curvature of the volume preserving diffeomorphism group of a Riemannian manifold $M$ can give information about the stability of inviscid, incompressible fluid flows on $M$. We demonstrate that the submanifold of the volumorphism group of the solid flat torus generated by axisymmetric fluid flows with swirl, denoted by $\mathcal{D}_{\mu,E}(M)$, has positive sectional curvature in every section containing the field $X=u(r)\partial_{\theta}$ iff $\partial_{r}(ru^{2})>0$. This is in sharp contrast to the situation on $\mathcal{D}_{\mu}(M)$, where only Killing fields $X$ have nonnegative sectional curvature in all sections containing it. We also show that this criterion guarantees the existence of conjugate points on $\mathcal{D}_{\mu,E}(M)$ along the geodesic defined by $X$.

###### Keywords
Axisymmetric, Euler equation, Ideal fluid, Curvature,Euler-Arnold, Stability

## 1 Introduction

Let $(M,g)$ be a Riemannian manifold of dimension at least two with Riemannian volume form $\mu$. The configuration space for inviscid, incompressible fluid flows on $M$ is the collection of smooth volume-preserving diffeomorphisms (volumorphisms) of $M$, denoted by $\mathcal{D}_{\mu}(M)$. [Arnold2014] showed in 1966 that flows obeying the Euler equations for inviscid, incompressible fluid flow can formally 11This was proved rigorously by [Ebin and Marsden1970], by working in the context of Sobolev $H^{s}$ diffeomorphisms for $s>\tfrac{1}{2}\dim{M}+1$. Here for simplicity we will work in the context of smooth diffeomorphisms since the curvature formulas are the same either way. be realized as geodesics on $\mathcal{D}_{\mu}(M)$. Using this framework, questions of fluid mechanics can be re-phrased in terms of the Riemannian geometry of $\mathcal{D}_{\mu}(M)$. An overview of this is given in [Arnold and Khesin1998] or more recently in [Khesin et al.2013]. Of particular interest is the sectional curvature of $\mathcal{D}_{\mu}(M)$. As in finite dimensional geometry, given two geodesics with varying initial velocities in a region of strictly positive (resp. negative) sectional curvature, the two geodesics will converge (resp. diverge) via the Rauch Comparison theorem. In terms of fluid mechanics, this corresponds to the Lagrangian stability (resp. instability) of the associated fluid flows.

Arnold showed that the sectional curvature $K(X,Y)$ of the plane in $T_{\text{id}}\mathcal{D}_{\mu}(M)$ spanned by $X$ and $Y$ is often negative but occasionally positive. [Rouchon1992] sharpened this to show that if $M\subset\mathbb{R}^{3}$, then $K(X,Y)\geq 0$ for every $Y\in T_{\text{id}}\mathcal{D}_{\mu}(M)$ if and only if $X$ is a Killing field (i.e., one for which the flow generates a family of isometries). This result was generalized by [Misiołek1993] and the second author ([Preston2002]) for any manifold with $\dim{M}\geq 2$. This gives the impression that, in general, $D_{\mu}(M)$ will mostly be negatively curved. The question of when one can expect a divergence free vector field to give nonpositive sectional curvature remains open. However, the second author ([Preston2005]) provided criteria for divergence free vector fields of the form $X=u(r)\partial_{\theta}$ on the area-preserving diffeomorphism groups of a rotationally-symmetric surface for which the sectional curvature $K(X,Y)$ is nonpositive for all $Y$.

Our goal in this paper is to extend the curvature computation to $\mathcal{D}_{\mu,E}(M)$, the group of volumorphisms commuting with the flow of a Killing field $E$. In particular, we consider the solid flat torus, $M=D^{2}\times S^{1}$, where $D^{2}$ is the unit disk in $\mathbb{R}^{2}$ and $S^{1}$ is the unit circle, with cylindrical coordinates $(r,\theta,z)$ for $0\leq r\leq 1$ and $\theta,z\in[0,2\pi]$. We may think of this more concretely as the subset of $\mathbb{R}^{3}$ with the planes $z=0$ and $z=2\pi$ identified, where $E=\partial_{\theta}$ is the field corresponding to rotation in the disc. Fluid flows on this manifold correspond to axisymmetric ideal flows with swirl on the solid infinite cylinder, which are $2\pi$-periodic in the $z$-direction. We consider steady fluid velocity fields of the form $X=u(r)\partial_{\theta}$. The submanifold $\mathcal{D}_{\mu,E}(M)$ is a totally geodesic submanifold of $\mathcal{D}_{\mu}(M)$ (see [Vizman1999], as well as [Haller et al.2002]; [Modin et al.2011] for the general situation in the smooth context, or see the preprint [Ebin and Preston2013] for the Sobolev diffeomorphism context), corresponding to the fact that an ideal fluid which is initially independent of $\theta$ will always remain so. Hence we compute sectional curvatures $K(X,Y)$ where $Y\in T_{\text{id}}\mathcal{D}_{\mu,E}(M)$ is divergence-free and axisymmetric, i.e., $[E,Y]=0$.

In [Preston2005] the second author effectively showed that when $X$ was considered as an element of $\mathcal{D}_{\mu,F}(M)$ where $F=\frac{\partial}{\partial z}$ (corresponding to considering $X$ as a two-dimensional flow rather than a three-dimensional flow), the sectional curvature satisfied $K(X,Y)\leq 0$ for every $Y\in T_{\text{id}}\mathcal{D}_{\mu,F}(M)$ regardless of $u(r)$. By contrast we show here that if $u$ satisfies the condition

 $\frac{d}{dr}\big(ru(r)^{2}\big)>0,$ (1)

then $K(X,Y)>0$ for every $Y\in T_{\text{id}}\mathcal{D}_{\mu,E}(M)$. We will also show that $\frac{d}{dr}\big(ru(r)^{2}\big)\geq 0$ implies that $K(X,Y)\geq 0$. This does not contradict the result of Rouchon, since the proof of that result relies on being able to construct a divergence-free velocity field with small support which points in a given direction and is orthogonal to another direction, and there are not enough divergence-free vector fields in the axisymmetric case to accomplish this here.

The fact that the curvature is strictly positive in every section containing $X$ makes it natural to ask whether there are conjugate points along every such corresponding geodesic. Unfortunately the Rauch comparison theorem cannot be used here, since $\inf_{Y\in T_{\text{id}}\mathcal{D}_{\mu,E}(M)}K(X,Y)=0$ even if (1) holds. Nonetheless we can show that as long as

 $ru(r)u^{\prime}(r)+2u(r)^{2}>0,$ (2)

the geodesic formed by $X=u(r)\partial_{\theta}$ has infinitely many monoconjugate points. It is easy to see that condition (1) implies (2). We do this by solving the Jacobi equation explicitly. As in [Ebin et al.2006], where the case $u(r)\equiv 1$ was considered, we can prove that these monoconjugate points have an epiconjugate point as a limit point, so that the differential of the exponential map is not even weakly Fredholm.

## 2 The Formula for Curvature

We first compute the curvature of $\mathcal{D}_{\mu,E}(M)$ by expanding in a Fourier series in $z$. Here all our vector fields and functions are smooth on the compact manifold $M$, so that convergence of the series will never be an issue, as in the original computations of [Arnold2014]. If desired one could do the same computations in the Sobolev $H^{s}$ context, with $s>5/2$, and treat the curvature operator as a continuous linear operator in $H^{s}$, as done by [Misiołek1993], but the final curvature formula is the same in either case. Our method here is similar to that of the second author in [Preston2005], where the computations were two-dimensional.

Notice first of all that any smooth vector field $Y$ which is tangent to $\mathcal{D}_{\mu,E}(M)$ at the identity must be divergence-free and must commute with $E=\frac{\partial}{\partial\theta}$. Therefore we can write in the form

 $Y(r,z)=-\frac{g_{z}(r,z)}{r}\,\partial_{r}+\frac{g_{r}(r,z)}{r}\,\partial_{z}+ f(r,z)\,\partial_{\theta},$ (3)

where $f(0,z)=g(0,z)=0$ and $g(1,z)$ is constant in $z$ (in order to be well-defined on the axis of symmetry and to have $Y$ tangent to the boundary $r=1$). We think of the term $-\frac{g_{z}}{r}\partial_{r}+\frac{g_{r}}{r}\partial_{z}$ as an analogue of the skew-gradient in two dimensions. We may express $Y$ in a Fourier series in $z$ as $Y(r,z)=\sum_{n\in\mathbb{Z}}Y_{n}(r,z)$ where

 $Y_{n}(r,z)=e^{inz}\left[-\frac{in}{r}g_{n}(r)\,\partial_{r}+\frac{g_{n}^{ \prime}(r)}{r}\,\partial_{z}+f_{n}(r)\,\partial_{\theta}\right].$ (4)

On any Riemannian manifold $(M,g)$ with volume form $\mu$, a formula for the curvature tensor on $\mathcal{D}_{\mu}(M)$ is given by

 $R(Y,X)X=P\left(\nabla_{Y}P(\nabla_{X}X)-\nabla_{X}P(\nabla_{Y}X)+\nabla_{[X,Y] }X\right),$ (5)

where $P(X)$ is the projection onto the divergence-free part of $X$. Concretely, $P(X)$ is obtained by solving the Neumann boundary value problem

 $\displaystyle\begin{cases}\Delta q=\mbox{div }X&\quad\mbox{ in }~{}M\\ \left<\nabla q,\overrightarrow{n}\right>=\left& \quad\mbox{ on }~{}\partial M\end{cases}$

for $q$ and then setting $P(X)=X-\nabla q$. The non-normalized sectional curvature is then given by

 $\overline{K}(X,Y)=\langle\!\langle R(Y,X)X,\overline{Y}\rangle\!\rangle=\int_{ M}\left\mu.$ (6)

See [Misiołek1993] for the derivation of the formula we use here. Our goal now is to compute $R(Y_{n},X)X$, and to do this we first need to compute $P(\nabla_{Y_{n}}X)$.

###### Lemma 1.

Suppose $Y_{n}$ is of the form (4) and $X=u(r)\,\partial_{\theta}$ in cylindrical coordinates on $M$. Then the covariant derivative $P(\nabla_{Y_{n}}X)$ in $\mathcal{D}_{\mu,E}(M)$ is given by $P(\nabla_{Y_{0}}X)=0$ and

 $\displaystyle P(\nabla_{Y_{n}}X)$ $\displaystyle=$ $\displaystyle\nabla_{Y_{n}}X-\nabla(q_{n}e^{inz})=-(rf_{n}u+q_{n}^{\prime})e^{ inz}\partial_{r}$ $\displaystyle-\frac{in}{r}g_{n}\left(u^{\prime}+\frac{u}{r}+rq_{n}\right)e^{ inz}\partial_{\theta}\quad{\rm for}~{}n\in\mathbb{N},$

where

 $q_{n}(r)=-\zeta_{n}(r)H_{n}(r)+\xi_{n}(r)J_{n}(r),$ (7)

with

 $H_{n}(r)=\int_{0}^{r}s^{2}f_{n}(s)u(s)\xi_{n}^{\prime}(s)\,ds\quad\text{and} \quad J_{n}(s)=-\int_{r}^{1}s^{2}f_{n}(s)u(s)\zeta_{n}^{\prime}(s)\,ds,$ (8)

and

 $\displaystyle\xi_{n}(r)=I_{0}(nr)\quad{\rm and}\quad\zeta_{n}(r)=\tfrac{K_{1}( n)}{I_{1}(n)}I_{0}(nr)+K_{0}(nr),$

with $I_{0}$ and $K_{0}$ denoting the modified Bessel functions of the first and second kinds.

###### Proof.

Consider the cases $n=0$ and $n\in\mathbb{N}$ separately. For $n=0$ we have

 $\displaystyle Y_{0}=\tfrac{1}{r}\,g_{0}^{\prime}(r)\partial_{z}+f_{0}(r) \partial_{\theta}$

and $\nabla_{Y_{0}}X=-rf_{0}(r)u(r)\partial_{r}$.

This is also the gradient of a function, and thus $P(\nabla_{Y_{0}}X)=0.$

Now for $n\neq 0$,

 $\displaystyle\nabla_{Y_{n}}X=-rf_{n}ue^{inz}\partial_{r}-\frac{in}{r}g_{n} \left(u^{\prime}+\frac{u}{r}\right)e^{inz}\partial_{\theta}.$

The solution $q_{n}(r)e^{inz}$ of

 $\begin{cases}\Delta(q_{n}(r)e^{inz})=\mbox{div}(\nabla_{Y_{n}}X)&\quad\mbox{in }M,\\ \left.\left<\nabla(q_{n}e^{inz}),\overrightarrow{n}\right>\right|_{\partial M} =\left.\left<\nabla_{Y_{n}}X,\overrightarrow{n}\right>\right|_{\partial M}& \quad\mbox{on }\partial M,\end{cases}$ (9)

must satisfy the ordinary differential equation

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