# Vortex solution in elliptic coordinates

Wladimir Lyra New Mexico State University, Department of Astronomy, PO Box
30001 MSC 4500, Las Cruces, NM 88001, USA

###### Abstract

Vortices (flows with closed elliptic streamlines) are exact nonlinear
solutions to the compressible Euler equation. In this contribution, we use
differential geometry to derive the transformations between Cartesian and
elliptic coordinates, and show that in elliptic coordinates a constant
vorticity flow reduces to $\dot{\mu}=0$ and $\dot{\nu}={\rm const}$ along the
streamline $\mu_{0}$ that matches the vortex eccentricity.

##  1 Introduction

Vortices are important for planet formation, theorized as favorable locations
for dust trapping (Barge & Sommeria, 1995). Crescent-shaped asymmetries have
been observed in sub-mm images of protoplanetary disks (van der Marel et al.,
2013), although their unambiguous identification vortices has been elusive. A
patch of constant vorticity follows the solution ${\bm{u}}=\Omega
y/\chi\hat{{\bm{x}}},-x\chi\hat{{\bm{y}}}$, where $(x,y)$ are the Cartesian
coordinates, $\Omega$ is a constant and $\chi=x/y>1$ is the vortex aspect
ratio. Given the elliptic streamlines, a solution in terms of elliptic
coordinates $(\mu,\nu)$ is desirable.

##  2 Elliptical coordinates

The orthogonal elliptical coordinate system is

$\displaystyle x$ $\displaystyle=$ $\displaystyle f\cosh\mu\cos\nu,$ (1)
$\displaystyle y$ $\displaystyle=$ $\displaystyle f\sinh\mu\sin\nu,$ (2)

where $f=a\epsilon$ is the focal length, $a$ the semi-major axis, and
$\epsilon$ the eccentricity. Constant $\mu$ define ellipses, constant $\nu$
define hyperbolae. The coordinates describe confocal ellipses: the focal
distance is constant, so changing $\mu$ changes not only the semimajor axis
but also the eccentricity.

###  2.1 Metric

The metric of this system is

$\displaystyle g_{ij}$ $\displaystyle=$
$\displaystyle\frac{\partial{x^{\alpha}}}{\partial{q^{i}}}\frac{\partial{x^{\beta}}}{\partial{q^{j}}}g_{\alpha\beta},$
(3) $\displaystyle=$ $\displaystyle
f^{2}\left(\sinh^{2}\mu+\sin^{2}\nu\right)\,\delta_{ij}.$

where ${\bm{x}}=(x,y)$ and ${\bm{q}}=(\mu,\nu)$ are Cartesian and elliptic
coordinates; $g_{\alpha\beta}=\delta_{\alpha\beta}$ is the metric of Cartesian
space. From this transformation, it follows that the scale factors are equal

$\displaystyle h_{\mu}$ $\displaystyle=$
$\displaystyle\sqrt{g_{\mu\mu}}=f\sqrt{\sinh^{2}\mu+\sin^{2}\nu},$ (4)
$\displaystyle h_{\nu}$ $\displaystyle=$
$\displaystyle\sqrt{g_{\nu\nu}}=f\sqrt{\sinh^{2}\mu+\sin^{2}\nu}.$ (5)

We hereafter use $h=h_{\mu}=h_{\nu}$. We also use the equivalent definition

$h=\frac{f}{\sqrt{2}}\sqrt{\cosh\,2\mu-\cos\,2\nu}.$ (6)

The derivatives with respect to the coordinates are

$\displaystyle\partial_{\mu}h$ $\displaystyle=$
$\displaystyle\frac{f^{2}}{2h}\,\sinh\,2\mu,$ (7)
$\displaystyle\partial_{\nu}h$ $\displaystyle=$
$\displaystyle\frac{f^{2}}{2h}\,\sin\,2\nu.$ (8)

We calculate the Christoffel symbols in non-coordinate basis

$\displaystyle\Gamma_{\hat{\alpha}\hat{\beta}\hat{\gamma}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(c_{\hat{\alpha}\hat{\beta}\hat{\gamma}}+c_{\hat{\alpha}\hat{\gamma}\hat{\beta}}-c_{\hat{\beta}\hat{\gamma}\hat{\alpha}}\right),$
(9) $\displaystyle\Gamma^{\hat{\alpha}}_{\hat{\beta}\hat{\gamma}}$
$\displaystyle=$ $\displaystyle
g^{\hat{\alpha}\hat{\zeta}}\Gamma_{\hat{\zeta}\hat{\beta}\hat{\gamma}},$ (10)

where
$c_{\hat{\beta}\hat{\gamma}\hat{\alpha}}=g_{\hat{\alpha}\hat{\zeta}}{c_{\hat{\beta}\hat{\gamma}}}^{\hat{\zeta}}$
are the connection coefficients, given by

$[e_{\hat{\beta}},e_{\hat{\gamma}}]={c_{\hat{\beta}\hat{\gamma}}}^{\hat{\alpha}}\partial_{\hat{\alpha}}.$
(11)

Given that $e_{\hat{\mu}}=h^{-1}\partial_{\mu}$ and
$e_{\hat{\nu}}=h^{-1}\partial_{\nu}$, we have

$\displaystyle[e_{\hat{\mu}},e_{\hat{\nu}}]$ $\displaystyle=$
$\displaystyle\frac{1}{h}\left[\frac{\partial{}}{\partial{\mu}}\left(\frac{1}{h}\frac{\partial{}}{\partial{\nu}}\right)-\frac{\partial{}}{\partial{\nu}}\left(\frac{1}{h}\frac{\partial{}}{\partial{\mu}}\right)\right]$
(12) $\displaystyle=$
$\displaystyle\frac{f^{2}}{2h^{4}}\left(\sin\,2\nu\frac{\partial{}}{\partial{\mu}}-\sinh\,2\mu\frac{\partial{}}{\partial{\nu}}\right)$
$\displaystyle=$
$\displaystyle\frac{f^{2}}{2h^{3}}\left(\sin\,2\nu\,\partial_{\hat{\mu}}-\sinh\,2\mu\,\partial_{\hat{\nu}}\right)$
$\displaystyle=$ $\displaystyle-[e_{\hat{\nu}},e_{\hat{\mu}}].$

The connection coefficients are thus

$\displaystyle{c_{{\hat{\mu}}{\hat{\nu}}}}^{\hat{\mu}}=-{c_{{\hat{\nu}}{\hat{\mu}}}}^{\hat{\mu}}$
$\displaystyle=$
$\displaystyle{\color[rgb]{1,1,1}-}\frac{f^{2}}{2h^{3}}\sin\,2\nu,$ (13)
$\displaystyle{c_{{\hat{\mu}}{\hat{\nu}}}}^{\hat{\nu}}=-{c_{{\hat{\nu}}{\hat{\mu}}}}^{\hat{\nu}}$
$\displaystyle=$ $\displaystyle-\frac{f^{2}}{2h^{3}}\sinh\,2\mu.$ (14)

And the Christoffel symbols are

$\displaystyle\Gamma^{\hat{\mu}}_{{\hat{\nu}}{\hat{\mu}}}=-\Gamma^{\hat{\nu}}_{{\hat{\mu}}{\hat{\mu}}}$
$\displaystyle=$ $\displaystyle\frac{f^{2}}{2h^{3}}\sin\,2\nu,$ (15)
$\displaystyle\Gamma^{\hat{\nu}}_{{\hat{\mu}}{\hat{\nu}}}=-\Gamma^{\hat{\mu}}_{{\hat{\nu}}{\hat{\nu}}}$
$\displaystyle=$ $\displaystyle\frac{f^{2}}{2h^{3}}\sinh\,2\mu.$ (16)

The elliptic and Cartesian unit vectors $\hat{e}_{i}$ and $\hat{x}_{i}$
transform according to

$\hat{e}_{i}=\frac{1}{h_{i}}\frac{\partial x_{j}}{\partial e_{i}}\hat{x_{j}},$
(17)

i.e.,

$\displaystyle\left[\begin{array}[]{c}\hat{\mu}\\\ \hat{\nu}\\\
\end{array}\right]$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccc}h_{\mu}^{-1}\partial_{\mu}x&{\color[rgb]{1,1,1}..}&h_{\mu}^{-1}\partial_{\mu}y\\\
h_{\nu}^{-1}\partial_{\nu}x&{\color[rgb]{1,1,1}..}&h_{\nu}^{-1}\partial_{\nu}y\\\
\end{array}\right]\left[\begin{array}[]{c}\hat{x}\\\ \hat{y}\\\
\end{array}\right]$ (24) $\displaystyle=$
$\displaystyle\frac{f}{h(\mu,\nu)}\left[\begin{array}[]{ccc}{\color[rgb]{1,1,1}-}\sinh\mu\cos\nu&{\color[rgb]{1,1,1}..}&\cosh\mu\sin\nu\\\
-\cosh\mu\sin\nu&{\color[rgb]{1,1,1}..}&\sinh\mu\cos\nu\\\
\end{array}\right]\left[\begin{array}[]{c}\hat{x}\\\ \hat{y}\\\
\end{array}\right]$ (29)

This can be written compactly as

$\hat{e}_{i}=E_{ij}\hat{x}_{j},$ (30)

where $E_{ij}$ is the elliptic rotation matrix. Its inverse is

$\displaystyle E^{-1}$ $\displaystyle=$
$\displaystyle\frac{f}{h}\left[\begin{array}[]{ccc}\sinh\mu\cos\nu&{\color[rgb]{1,1,1}..}&-\cosh\mu\sin\nu\\\
\cosh\mu\sin\nu&{\color[rgb]{1,1,1}..}&{\color[rgb]{1,1,1}-}\sinh\mu\cos\nu\\\
\end{array}\right].$ (33)

The velocity is

$u_{i}=h_{i}\dot{q}_{i}\hat{q}_{i},$ (34)

which means

$\displaystyle{\bm{u}}$ $\displaystyle=$
$\displaystyle\dot{x}\hat{x}+\dot{y}\hat{y},$ (35) $\displaystyle=$
$\displaystyle h_{\mu}\dot{\mu}\hat{\mu}+h_{\nu}\dot{\nu}\hat{\nu}.$

We can also get the velocity by the rotation matrix

$u_{{\hat{e}}_{i}}=E_{ij}u_{c_{j}},$ (36)

i.e.,

$\displaystyle u_{\hat{\mu}}=E_{11}u_{x}+E_{12}u_{y},$ (37) $\displaystyle
u_{\hat{\nu}}=E_{21}u_{x}+E_{22}u_{y}.$ (38)

Yielding the variation of the coordinate bases

$\displaystyle\dot{\mu}$ $\displaystyle=$ $\displaystyle
fh^{-2}\left({\color[rgb]{1,1,1}-}\sinh\mu\cos\nu\,u_{x}+\cosh\mu\sin\nu\,u_{y}\right),$
(39) $\displaystyle\dot{\nu}$ $\displaystyle=$ $\displaystyle
fh^{-2}\left(-\cosh\mu\sin\nu\,u_{x}+\sinh\mu\cos\nu\,u_{y}\right).$ (40)

##  3 Vortex motion

Consider a vortex in Cartesian coordinates

$\displaystyle u_{x}$ $\displaystyle=$ $\displaystyle-\varOmega y\chi$ (41)
$\displaystyle u_{y}$ $\displaystyle=$
$\displaystyle{\color[rgb]{1,1,1}-}\varOmega x/\chi$ (42)

We seek to transform this into elliptic coordinates. The vortex motion occurs
on ellipses of constant eccentricity, whereas the elliptic coordinate system
defines confocal ellipses of different eccentricity. An elliptic coordinate
system based on constant eccentricity (Chang & Oishi, 2010), although matching
the flow geometry, is not orthogonal, which complicates analysis (Lyra & Lin,
2013). If the streamlines matched the eccentricities of the confocal ellipses,
the velocity would everywhere reduce to $\dot{\mu}=0$ and $\dot{\nu}={\rm
const}$. However, that is not the case, as one can verify that this is not
divergenceless. In fact, there is only one streamline that obeys $\dot{\mu}=0$
and $\dot{\nu}={\rm const}$, which is the streamline of eccentricity matching
the eccentricity of the vortex. This is the particular ellipse $\mu_{0}$,
given by $\tanh\mu_{0}=\chi^{-1}$. We write the velocities as

$\displaystyle u_{x}$ $\displaystyle=$
$\displaystyle-\varOmega\,f\,\cosh\mu\sin\nu,$ (43) $\displaystyle u_{y}$
$\displaystyle=$
$\displaystyle{\color[rgb]{1,1,1}-}\varOmega\,f\,\sinh\mu\cos\nu.$ (44)

We transform these into elliptical coordinates by the rotation matrix

$u_{{\hat{e}}_{i}}=E_{ij}u_{c_{j}}$ (45)

yielding

$\displaystyle u_{\hat{\mu}}$ $\displaystyle=$
$\displaystyle-\Omega\frac{f^{2}}{2h}\left(\frac{\cosh\,2\mu-\cosh\,2\mu_{0}}{\sinh\,2\mu_{0}}\right)\sin\,2\nu,$
(46) $\displaystyle u_{\hat{\nu}}$ $\displaystyle=$
$\displaystyle\Omega\frac{f^{2}}{2h}\left(\frac{\sinh\,2\mu}{\sinh\,2\mu_{0}}\right)(\cosh\,2\mu_{0}-\cos\,2\nu).$
(47)

The divergence is

$\displaystyle{\bm{\nabla}}\cdot{A}$ $\displaystyle=$ $\displaystyle
u^{\hat{\alpha}}_{;\hat{\alpha}}=u^{\hat{\alpha}}_{,\hat{\alpha}}+\Gamma^{\hat{\alpha}}_{\hat{\beta}\hat{\alpha}}u^{\hat{\beta}},$
(49) $\displaystyle=$ $\displaystyle
u^{\hat{\alpha}}_{,\hat{\alpha}}+\frac{f^{2}}{2h^{3}}\left(u^{\hat{\mu}}\sin\,2\nu+u^{\hat{\nu}}\sinh\,2\mu\right),$

or, abandoning the co-variant formulation,

${\bm{\nabla}}\cdot{A}=\frac{1}{h^{2}}\left(\frac{\partial{hu_{\hat{\mu}}}}{\partial{\mu}}+\frac{\partial{hu_{\hat{\nu}}}}{\partial{\nu}}\right).$
(51)

we conclude that the flow is divergenceless.

Eq. (46) and Eq. (47) may seem daunting at first, but following the motion at
the ellipse of $\mu$=$\mu_{0}$ simplifies it considerably. For $\mu=\mu_{0}$,
Eq. (46) cancels. For Eq. (47), the factor in parentheses becomes unity; the
next term, given Eq. (6), is $2h/f^{2}$. Thus, for $\mu$=$\mu_{0}$, the motion
is $u_{\hat{\mu}}$=0, $u_{\hat{\nu}}$=$\varOmega\,h_{0}$. Comparing with Eq.
(35) yields

$\displaystyle\dot{\mu}$ $\displaystyle=$ $\displaystyle 0,$ (52)
$\displaystyle\dot{\nu}$ $\displaystyle=$ $\displaystyle\varOmega.$ (53)

For the particular $\mu=\mu_{0}$ ellipse, the motion has constant $\mu$: a
closed elliptic streamline. The angle $\nu$ rotates uniformly. Notice that
this does not mean that the velocity itself is uniform, since $h$ depends on
$\nu$. The explicit dependency of $u_{\nu}$ on $\nu$ is

$\displaystyle u_{\nu}^{2}$ $\displaystyle=$
$\displaystyle\frac{\varOmega^{2}\,f^{2}}{2}\left(\cosh\,2\mu-\cos\,2v\right)$
(54) $\displaystyle=$
$\displaystyle\varOmega^{2}\,f^{2}\left(\sinh^{2}\mu+\sin^{2}\,v\right).$

##  4 Energy conservation

That the kinetic energy $K=u_{\nu}^{2}/2$ depends on $\nu$, a function of
time, may seem strange at first. We show that this happens because the
velocity change is compensated by a change in pressure ($p$), conserving the
total energy. The energy equation is

$\frac{\partial E}{\partial
t}=-{\bm{\nabla}}\cdot{\left[{\bm{u}}\left(E+p\right)\right]}+{\bm{F}}_{b}\cdot{\bm{u}}$
(55)

where ${\bm{F}}_{b}$ is a body force. The total energy is $E=K+\varepsilon$,
where and $\varepsilon=k_{B}T$ is the internal energy ($k_{B}$ is Boltzmann’s
constant and $T$ is the temperature). In the absence of a body force and for
constant temperature, this reduces to

$\frac{\partial}{\partial
t}\left(\frac{u^{2}}{2}\right)=-\left({\bm{u}}\cdot{\bm{\nabla}}\right)\left(\frac{u^{2}}{2}+p/\rho\right)$
(56)

therefore

$\frac{d}{dt}\left(\frac{u^{2}}{2}\right)=-\left({\bm{u}}\cdot{\bm{\nabla}}\right)p/\rho$
(57)

The enthalpy is found by Euler’s equation

$\displaystyle\partial_{x}p/\rho$ $\displaystyle=$ $\displaystyle-
u_{x}\partial_{x}u_{y}=\varOmega^{2}x$ (58) $\displaystyle\partial_{y}p/\rho$
$\displaystyle=$ $\displaystyle-u_{y}\partial_{y}u_{x}=\varOmega^{2}y$ (59)

Taking the $x$-derivative above and the $y$ below, we find
$\nabla^{2}{p/\rho}=\varOmega^{2}$; therefore

$p/\rho=\frac{1}{2}\varOmega^{2}\left(x^{2}+y^{2}\right)+C$ (60)

which is an intriguing result: an incompressible elliptical vortex produces an
axis-symmetric pressure distribution. Transforming into elliptic coordinates
and eliminating the constant

$p/\rho=\frac{1}{2}\Omega^{2}f^{2}(\cosh^{2}\mu\cos^{2}\nu+\sin^{2}\mu\sin^{2}\nu).$
(61)

Along the $\mu_{0}$ streamline, the advection reduces to the $\nu$-term

$\displaystyle-\left({\bm{u}}\cdot{\bm{\nabla}}\right)P/\rho$ $\displaystyle=$
$\displaystyle-u_{\nu}h^{-1}\partial_{\nu}p/\rho$ (62) $\displaystyle=$
$\displaystyle\frac{1}{2}\varOmega^{3}f^{2}\sin\,2\nu,$

whereas the time derivative of the kinetic energy is

$\displaystyle\frac{d}{dt}\left(\frac{u^{2}}{2}\right)$ $\displaystyle=$
$\displaystyle\Omega^{2}h\frac{dh}{dt}$ (63) $\displaystyle=$
$\displaystyle\frac{1}{2}\Omega^{3}f^{2}\sin\,2\nu.$

That the two variations match amounts to conservation of energy: along the
ellipse, the material slows down or speeds up in order to match the pressure
variation.

##  5 Euler equation in elliptical coordinates

We consider now the force balance. We use the transformations here derived to
write the Euler equation in elliptic coordinates

$\partial_{t}{\bm{u}}+\left({\bm{u}}\cdot{\bm{\nabla}}\right){\bm{u}}=-{\bm{\nabla}}{p/\rho}.$
(64)

Using covariant derivatives, this reads

$\partial_{t}u^{\hat{k}}+u^{\hat{p}}\partial_{\hat{p}}u^{\hat{k}}+\Gamma^{\hat{k}}_{\hat{m}\hat{n}}u^{\hat{m}}u^{\hat{n}}=-\partial_{\hat{k}}{p/\rho}.$
(65)

For $\mu$, the correction due to the Christoffel symbols is

$\displaystyle\Gamma^{\hat{\mu}}_{{\hat{m}}{\hat{n}}}u^{{\hat{m}}}u^{{\hat{n}}}$
$\displaystyle=$ $\displaystyle\Gamma^{{\hat{\mu}}}_{{\hat{\nu}}{\hat{\mu}}},$
(66) $\displaystyle=$
$\displaystyle\frac{f^{2}}{2h^{3}}\left[u^{{\hat{\nu}}}u^{{\hat{\mu}}}\,\sin\,2\nu-(u^{{\hat{\nu}}})^{2}\sinh\,2\mu\right].$

The same procedure for $\nu$ yields

$\displaystyle\Gamma^{{\hat{\nu}}}_{{\hat{m}}{\hat{n}}}u^{{\hat{m}}}u^{{\hat{\nu}}}$
$\displaystyle=$
$\displaystyle\Gamma^{\hat{\nu}}_{{\hat{\nu}}{\hat{\mu}}}u^{\hat{\mu}}u^{\hat{\nu}}+\Gamma^{\hat{\mu}}_{{\hat{\mu}}{\hat{\mu}}}(u^{{\hat{\mu}}})^{2},$
(67) $\displaystyle=$
$\displaystyle\frac{f^{2}}{2h^{3}}\left[u^{\hat{\nu}}u^{\hat{\mu}}\,\sinh\,2\mu-(u^{\hat{\mu}})^{2}\,\sin\,2\nu\right].$

Abandoning the co-variant notation

$\displaystyle\partial_{t}\,u_{\mu}$ $\displaystyle=$
$\displaystyle-\left(h^{-1}u_{\mu}\partial_{\mu}+h^{-1}u_{\nu}\partial_{\nu}\right)u_{\mu}-h^{-1}\partial_{\mu}{p/\rho}-\frac{f^{2}}{2h^{3}}\left(u_{\nu}u_{\mu}\,\sin\,2\nu-
u_{\nu}^{2}\sinh\,2\mu\right),$ (68) $\displaystyle\partial_{t}\,u_{\nu}$
$\displaystyle=$
$\displaystyle-\left(h^{-1}u_{\mu}\partial_{\mu}+h^{-1}u_{\nu}\partial_{\nu}\right)u_{\nu}-h^{-1}\partial_{\nu}{p/\rho}-\frac{f^{2}}{2h^{3}}\left(u_{\nu}u_{\mu}\,\sinh\,2\mu-
u_{\mu}^{2}\,\sin\,2\nu\right).$ (69)

This differs from the usual equations by the presence of the extra force

$\displaystyle{\bm{F}}$ $\displaystyle=$
$\displaystyle\frac{f^{2}}{2h^{3}}\left[\left(u_{\nu}u_{\mu}\,\sin\,2\nu-
u_{\nu}^{2}\sinh\,2\mu\right)\hat{\mu}+\left(u_{\nu}u_{\mu}\,\sinh\,2\mu-
u_{\mu}^{2}\,\sin\,2\nu\right)\hat{\nu}\right]$ (70)

Contracting this force with the velocity yields ${\bm{F}}\cdot{\bm{u}}=0$,
which shows that this force is inertial. For the vortical flow, again
following the $\mu_{0}$ streamline where $\dot{\mu}=0$ and
$\dot{\nu}=\varOmega$, these reduce to

$\displaystyle-\frac{\varOmega^{2}f^{2}}{2}\sinh\,2\mu_{0}$ $\displaystyle=$
$\displaystyle-\partial_{\mu}{p/\rho}$ (71)
$\displaystyle\left(u_{\nu}\partial_{\nu}\right)u_{\nu}$ $\displaystyle=$
$\displaystyle-\partial_{\nu}{p/\rho}$ (72)

I.e, a constant centrifugal force that balances the normal pressure gradient,
and inertia in the tangential direction exchanging kinetic energy with the
pressure field. Fig 1 sketches the forces.

Figure 1: Force balance in an elliptic vortex streamline (solid line). Dotted
circles represent the pressure contours. The velocity (black arrow) is tangent
to the streamline, in the $\hat{\nu}$ direction. The pressure gradient (blue
arrow) is broken down in its $\hat{\mu}$ and $\hat{\nu}$ components (brown
arrows). The $\hat{\mu}$ component is balanced by the centrifugal force (red
arrow); the $\hat{\nu}$ component is balanced by advection.

## References

  * Barge & Sommeria (1995) Barge, P., & Sommeria, J. 1995, A&A, 295, L1
  * Chang & Oishi (2010) Chang, P., & Oishi, J. S. 2010, ApJ, 721, 1593, doi: 10.1088/0004-637X/721/2/1593
  * Lyra & Lin (2013) Lyra, W., & Lin, M.-K. 2013, ApJ, 775, 17, doi: 10.1088/0004-637X/775/1/17
  * van der Marel et al. (2013) van der Marel, N., van Dishoeck, E. F., Bruderer, S., et al. 2013, Science, 340, 1199, doi: 10.1126/science.1236770