Time Evolution of Concentrated Vortex Rings

Abstract

We study the time evolution of an incompressible fluid with axisymmetry without swirl when the vorticity is sharply concentrated. In particular, we consider N disjoint vortex rings of size $\varepsilon$ and intensity of the order of $|\log \varepsilon |^{ -1}$. We show that in the limit $\varepsilon \rightarrow 0$, when the density of vorticity becomes very large, the movement of each vortex ring converges to a simple translation, at least for a small but positive time.

Appendices

Appendix A. Planar Symmetry

In this appendix we briefly recall some results concerning the time evolution of concentrated Euler flows with planar symmetry (without giving a complete list of references on this topic). The Euler equations for an incompressible inviscid fluid in the whole space with planar symmetry and constant density reads

$$\begin{aligned} \partial _t\omega (x,t) + (u \cdot \nabla ) \omega (x,t) = 0, \quad \nabla \cdot u(x,t) = 0,\quad x\in {{\mathbb {R}}}^2. \end{aligned}$$

By assuming that u vanishes at infinity, the velocity field is reconstructed from the vorticity as

$$\begin{aligned} u(x,t) = \int {\mathrm {d}}y\, K(x-y) \, \omega (y,t), \end{aligned}$$

with $K(\cdot )$ as in Eq. (3.7).

We assume that initially the vorticity is concentrated in N blobs of the form

$$\begin{aligned} \omega _\varepsilon (x,0) = \sum _{i=1}^N \omega _{i,\varepsilon }(x,0), \end{aligned}$$

where $\omega _{i,\varepsilon }(x,0)$, $i=1,\ldots , N$, are functions with definite sign such that

$$\begin{aligned} \Lambda _{i,\varepsilon }(0) := \mathop {\mathrm{supp}}\nolimits \, \omega _{i,\varepsilon }(\cdot ,0) \subset \Sigma (z_i|\varepsilon ), \quad \Sigma (z_i|\varepsilon )\cap \Sigma (z_j|\varepsilon )=\emptyset \quad \forall \, i \ne j, \end{aligned}$$

with $\varepsilon \in (0,1)$ a small parameter and the points $z_i\in {{\mathbb {R}}}^2$.

In this case, the solution of the Euler equations is strictly related to the point vortex model, the dynamical system defined by the following system of ordinary differential equations,

$$\begin{aligned} {\dot{z}}_i(t) = \sum _{\begin{array}{c} j=1 \\ j\ne i \end{array}}^N A_j K(z_i(t)-z_j (t)), \end{aligned}$$

(A.1)

where $A_j$ is called the “intensity” of the jth vortex. More precisely, it has been proven that in general, for small $\varepsilon$, the time evolution of the vorticity has the same form,

$$\begin{aligned} \Lambda _{i,\varepsilon }(t) := \mathop {\mathrm{supp}}\nolimits \, \omega _{i,\varepsilon }(\cdot ,r_\varepsilon (t)) \subset \Sigma (z_i(t),r_\varepsilon (t)), \end{aligned}$$

with

$$\begin{aligned} \Sigma (z_i(t),r_\varepsilon (t)) \cap \Sigma (z_j(t),r_\varepsilon (t)) =0\quad \forall \, i \ne j\;, \end{aligned}$$

where $r_\varepsilon (t)$ is a positive function and $\{z_i(t); i=1,\ldots , N\}$ is the solution to Eq. (A.1) with initial conditions $z_i(0)=z_i$ and intensity $A_i = \int {\mathrm {d}}x \, \omega _{i,\varepsilon }(x,0)$. The point $z_i(t)$ in the plane thus identifies a straight line in the space around which the vorticity is concentrated.

When all the intensities $A_i$ have the same sign (positive or negative) Eq. (A.1) has a global in time solution. Instead, if the signs are different there are examples in which collapses could occur (e.g., two vortices collide or a vortex goes to infinity in finite time). However, these pathological events are exceptional. For a review on this issue see, for instance, [16].

The point vortex model has been introduced in the eighteenth century by Helmholtz as a “solution” of the Euler equations, and widely analyzed in many papers. One hundred years later, it has been considered as a numerical approximation of the Euler evolution for very irregular initial data. As a numerical tool, this system is considered when N is very large and the intensity of each vortex very small (of the order of $N^{-1}$). On this topic there are several papers, we only quote the recent review [5].

As explained at the beginning of this appendix, a different point of view is adopted for finite N and consists in considering the point vortex model as an approximation of N very concentrated vortices, say with support of diameter 2$\varepsilon \rightarrow 0$. It is worthwhile to emphasize that it cannot be an approximation of each evolved path, because the length of the trajectory of a fluid element diverges as $\varepsilon \rightarrow 0$. On the other hand, by virtue of rapid rotations, a system of N disjoint concentrated patches of vorticity converges as a measure to a linear combination of Dirac measures $\sum _{i=1}^N A_i \delta _{z_i(t)}$ for any positive time. This convergence has been proven 25 years ago. Recently, the problem on how long the sharp localization of the vorticity remains valid has been analyzed in [4].

It is possible to introduce a small viscosity $\nu$ and study the small viscosity limit $\nu \rightarrow 0$. For finite times the status of art has been already discussed in Sect. 4.2. The validity of the convergence on very long times (diverging as $\varepsilon \rightarrow 0$) is analyzed in [6].

Appendix B. Proof of Some Technical Results

Proof of Lemma 3.1

The proof of [2, Lemma 2.1] is based on the conservation along the motion of the kinetic energy,

$$\begin{aligned} E = \frac{1}{2} \int {\mathrm {d}}{{\varvec{\xi }}}\, |{{\varvec{u}}} ({{\varvec{\xi }}},t)|^2 = \frac{1}{2} \int {\mathrm {d}}z \int _0^\infty {\mathrm {d}}r \, 2\pi r \big [u_z(z,r,t)^2+u_r(z,r,t)^2\big ]. \end{aligned}$$

More precisely, the assumptions on the initial vorticity, together with Eq. (1.9) and the conservation of the quantities

$$\begin{aligned} M_0 = \int {\mathrm {d}}z \int _0^\infty {\mathrm {d}}r \, \omega _\varepsilon (z,r,t), \quad M_2 = \int {\mathrm {d}}z \int _0^\infty {\mathrm {d}}r \, r^2 \omega _\varepsilon (z,r,t) , \end{aligned}$$

allow to compute the asymptotic behavior as $\varepsilon \rightarrow 0$ of the energy E, from which the desired concentration estimate is deduced.

In this case, since the vector field ${{\varvec{F}}}^\varepsilon = (F^\varepsilon _z,F^\varepsilon _r,F^\varepsilon _\theta ) := (F^\varepsilon _1,F^\varepsilon _2,0)$ has zero divergence, the conservation laws of the energy E and of $M_0$ are still valid. Indeed,³

$$\begin{aligned} \begin{aligned} \dot{E}&= \int {\mathrm {d}}{{\varvec{\xi }}}\, {{\varvec{u}}} \cdot \partial _t {{\varvec{u}}} = - \int {\mathrm {d}}{{\varvec{\xi }}}\, {{\varvec{u}}} \cdot \big [({{\varvec{u}}} + {{\varvec{F}}}^\varepsilon )\cdot \nabla {{\varvec{u}}} + \nabla p\big ] \\&= \int {\mathrm {d}}{{\varvec{\xi }}}\, \bigg \{\frac{|{{\varvec{u}}}|^2}{2} \nabla \cdot ({{\varvec{u}}} + {{\varvec{F}}}^\varepsilon ) + p\, \nabla \cdot {{\varvec{u}}} \bigg \} = 0, \end{aligned} \end{aligned}$$

while, by Liouville’s theorem and Eq. (1.9),

$$\begin{aligned} M_0(t) = \frac{1}{2\pi } \int {\mathrm {d}}{{\varvec{\xi }}}\, \frac{\omega _\varepsilon ({{\varvec{\xi }}},t)}{r} = \int {\mathrm {d}}{{\varvec{\xi }}}_0 \, \frac{\omega _\varepsilon ({{\varvec{\xi }}}_0,0)}{r_0} = M_0(0) \end{aligned}$$

(B.1)

(above, the coordinate transformation is ${{\varvec{\xi }}}=\phi ^t({{\varvec{\xi }}}_0)$ with $\phi ^t$ the flow generated by $\dot{{{\varvec{\xi }}}} = {{\varvec{u}}}({{\varvec{\xi }}},t) + {{\varvec{F}}}^\varepsilon ({{\varvec{\xi }}},t)$).

Concerning the variation of $M_2$, since $\omega _\varepsilon (z,r,t)$ has compact support, we can apply Eq. (2.10) with $f(x,t) = x_2^2$, so that

$$\begin{aligned} \dot{M}_2 = \int {\mathrm {d}}x \, \omega _\varepsilon (x,t) \,2 x_2 F^\varepsilon _2(x,t) , \end{aligned}$$

which implies $|\dot{M}_2| \le 2C_F |\log \varepsilon |^{-3/2} \sqrt{M_2}$ in view of Assumption 2.1, item (b). Therefore, $M_2 \le 2(|{\bar{\zeta }}_2| + \varepsilon )^2 M_0 + 4C_F^2T^2 |\log \varepsilon |^{-3} \le {({\mathrm{const.}})\,}|\log \varepsilon |^{-1}$. This is the same estimate, but for a larger constant, which is obtained in absence of $F^\varepsilon$. Since the particular value of this constant is easily seen to be irrelevant in the proof of [2, Thm. 1], the lemma follows. $\square$

Proof of Lemma 3.3

Letting

$$\begin{aligned} a(x,y) := \frac{|x-y|}{\sqrt{x_2y_2}}, \end{aligned}$$

(B.2)

we have,

$$\begin{aligned} H(x,y) = - \frac{I_1(a(x,y))}{2\pi } \frac{(x-y)^\perp }{x_2\sqrt{x_2y_2}} + \frac{I_2(a(x,y))}{2\pi }\frac{1}{x_2}\sqrt{\frac{y_2}{x_2}} \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \end{aligned}$$

where

$$\begin{aligned} I_1(a) = \int _0^\pi {\mathrm {d}}\theta \, \frac{\cos \theta }{[a^2+2(1-\cos \theta )]^{3/2}} , \quad I_2(a) = \int _0^\pi {\mathrm {d}}\theta \, \frac{1-\cos \theta }{[a^2+2(1-\cos \theta )]^{3/2}}. \end{aligned}$$

By an explicit computation, see, e.g., the Appendix in [12], for any $a>0$,

$$\begin{aligned} I_1(a) = \frac{1}{a^2} + \frac{1}{4} \log \frac{a}{1+a} + \frac{c_1(a)}{1+a}, \quad I_2(a) = -\frac{1}{2} \log \frac{a}{1+a} + \frac{c_2(a)}{1+a}, \end{aligned}$$

with $c_1(a)$, $c_1'(a)$, $c_2(a)$, $c_2'(a)$ uniformly bounded for $a\in (0,+\infty )$. Therefore, the kernel R(x, y) defined by (3.8) is given by

$$\begin{aligned} {{\mathcal {R}}} (x,y) = \sum _{j=1}^6 R^j(x,y), \end{aligned}$$

with, for $a = a(x,y)$ as in (B.2),

$$\begin{aligned} R^1(x,y)&= \frac{1}{2\pi } \bigg (1 - \sqrt{\frac{y_2}{x_2}}\bigg ) \frac{(x-y)^\perp }{|x-y|^2},\;\; R^2(x,y) = \frac{1}{8\pi } \bigg (\log \frac{1+a}{a}\bigg ) \frac{(x-y)^\perp }{x_2\sqrt{x_2y_2}}, \\ R^3(x,y)&= \frac{1}{4\pi x_2} \sqrt{\frac{y_2}{x_2}} \bigg (\log \frac{|x-y|}{1+|x-y|} - \log \frac{a}{1+a}\bigg ) \begin{pmatrix} 1 \\ 0 \end{pmatrix} , \\ R^4(x,y)&= \frac{1}{4\pi x_2} \bigg (1-\sqrt{\frac{y_2}{x_2}}\bigg ) \log \frac{|x-y|}{1+|x-y|} \begin{pmatrix} 1 \\ 0 \end{pmatrix} , \\ R^5(x,y)&= -\frac{c_1(a)}{2\pi (1+a)} \frac{(x-y)^\perp }{x_2\sqrt{x_2y_2}}, \quad R^6(x,y) = \frac{c_2(a)}{2\pi (1+a)x_2}\sqrt{\frac{y_2}{x_2}} \begin{pmatrix} 1 \\ 0 \end{pmatrix}. \end{aligned}$$

Using that

$$\begin{aligned} \bigg |1-\sqrt{\frac{y_2}{x_2}} \bigg | = \frac{|y_2-x_2|}{x_2+\sqrt{x_2y_2}} \le \frac{|x-y|}{x_2} \end{aligned}$$

and

$$\begin{aligned} \bigg |\log \frac{|x-y|}{1+|x-y|} - \log \frac{a}{1+a}\bigg | = \bigg | \log \frac{1+a}{(x_2y_2)^{-1/2}+a}\bigg | \le \frac{1}{2} |\log (x_2y_2)|, \end{aligned}$$

we have,

$$\begin{aligned}&|R^1(x,y)| = \frac{1}{2\pi } \bigg |1-\sqrt{\frac{y_2}{x_2}} \bigg | \frac{1}{|x-y|}\le \frac{1}{2\pi x_2}, \\&|R^2(x,y)| = \frac{1}{8\pi x_2} \bigg (\log \frac{1+a}{a}\bigg ) \frac{|x-y|}{\sqrt{x_2y_2}} \le \frac{1}{8\pi x_2} \, \sup _{\rho>0}\bigg ( \rho \log \frac{1+\rho }{\rho }\bigg ), \\&|R^3(x,y)| + |R^6(x,y)| \le \frac{1}{4\pi x_2} \sqrt{\frac{y_2}{x_2}} \bigg (|\log (x_2y_2)| + \sup _{\rho>0} \frac{2c_2(\rho )}{1+\rho }\bigg ), \\&|R^4(x,y)| = \frac{1}{4\pi x_2} \bigg |1-\sqrt{\frac{y_2}{x_2}} \bigg | \log \frac{1+|x-y|}{|x-y|} \le \frac{1}{4\pi x_2^2} \,\sup _{\rho>0}\bigg ( \rho \log \frac{1+\rho }{\rho }\bigg ), \\&|R^5(x,y)| = \frac{|c_1(a)|}{2\pi (1+a)} \frac{|x-y|}{x_2\sqrt{x_2y_2}} \le \frac{1}{2\pi x_2} \, \sup _{\rho >0}\frac{\rho c_1(\rho )}{1+\rho }. \end{aligned}$$

In conclusion, there is $C_0>0$ such that

$$\begin{aligned} \begin{aligned}&|R^1(x,y)| + |R^2(x,y)| + |R^5(x,y)| \le \frac{C_0}{x_2}, \quad |R^4(x,y)| \le \frac{C_0}{x_2^2}, \\&|R^3(x,y)| + |R^6(x,y)| \le \frac{C_0}{x_2} \sqrt{\frac{y_2}{x_2}} \bigg (1+|\log (x_2y_2)|\bigg ). \end{aligned} \end{aligned}$$

The lemma is thus proven. $\square$