# Slender vortex filaments in the Boussinesq Approximation

Marie Rodal<EMAIL_ADDRESS>FB Mathematik und Informatik, Freie
Universität Berlin, Arnimallee 6, 14195 Berlin, Germany  PLECO, Department of
Biology, University of Antwerp, Universiteitsplein 1 2610 Wilrijk, Belgium
Daniel Margerit  39 avenue des grands pins, 31660 Buzet-sur-Tarn, France
Rupert Klein  FB Mathematik und Informatik, Freie Universität Berlin,
Arnimallee 6, 14195 Berlin, Germany

###### Abstract

A model for the motion of slender vortex filaments is extended to include the
effect of gravity. The model, initially introduced by Callegari and Ting
(1978), is based on a matched asymptotic expansion in which the outer
solution, given by the Biot-Savart law, is matched with the inner solution
derived from the Navier-Stokes equations. Building on recent work by
Harikrishnan  _et al._ (2023) the Boussinesq approximation is applied such
that the density variations only enter in the gravity term. However, unlike
Harikrishnan  _et al._ (2023) the density variation enters at a lower order in
the asymptotic expansion, and thus has a more significant impact on the self-
induced velocity of the vortex filament. In this regime, which corresponds to
the regime studied by Chang and Smith (2018), the effect of gravity is given
by an alteration of the core constant, which couples the motion of the
filament to the motion within the vortical core, in addition to a change in
the compatability conditions (evolution equations) which determine the leading
order azimuthal and tangential velocity fields in the vortex core. The results
are used to explain certain properties of bouyant vortex rings, as well as
qualitatively explore the impact of gravity on tornado type atmospheric
vorticies.

##  I Introduction

Vorticies are a fundamental part of fluid dynamics, from small scale
turbulence to large scale phenomena such as hurricanes. The study of vorticies
and vortex motion is not only of theoretical interest, but of extreme
practical importance as vorticies like hurricanes and tornados result in
untold amounts of damages every year. Tornadoes, in particular, are difficult
to both predict and study due to their in general sudden appearance and
relatively short life span; tornadoes usually last between a few minutes to an
hour depending on tornado strength. Hence, there are considerable gaps in our
knowledge of the formation and general motion of vorticies of this type; for a
relatively recent review on the current understanding of tornadogenesis within
supercell thunderstorms we refer the reader to Markowski and Richardson (2009)
or Davies-Jones (2015). Despite the difficulties, considerable progress has
been made in constructing analytical models for describing specific aspects of
tornado motion; see Lewellen (1993) or Varaksin (2017) for a review. Yih
(2007) developed a simple model based on the Bernoulli equation, and used it
to demonstrate certain properties of tornado-like vorticies such as funnel
broadening and an increase in the azimuthal velocity as one approaches the
ground. Shtern, Borissov, and Fazle (1998) used a generalized model of the
planar vortex sink to explain bulge formation in tornados, where the tornado
funnel suddenly expands at some height above ground. Loper (2020) developed a
model for turbulent boundary layer flows beneath a vortex, in an attempt to
gain a better understanding of the mean flow near the ground in tornado-like
vorticies. The work of Loper complements earlier work by Oruba, Davidson, and
Dormy (2018), where the authors looked at eye formation in tropical cyclones.
We wish to approach the problem of tornado type atmospheric vorticies from a
slightly different angle, employing the tools developed for the modeling of
slender vortex filaments. While tropical cyclones can have a diameter of
several hundred kilometers, tornadoes are much narrower; typically on the
order of 100 meters, which, given that they can have a height of several
kilometers, puts them firmly in the category of slender vortex filaments.
The main difficulty in the modelling of such vorticies is that the Biot-Savart
law, which gives the induced velocity at any point outside of the filament,
diverges when the point of evaluation is on the filament itself. To overcome
this difficulty several approaches have been proposed, of which the local
induction approximation (LIA) is perhaps the most well known (e.g. Batchelor
(2000)). Another well known approach is the cut-off method, in which the
singularity is artificially removed by introducing a lower integration limit
(Saffman, 1970). Neither of these methods capture the dynamics of the vortex
core, resulting in potentially large modelling and predictive errors (Klein
and Knio, 1995). In an attempt to overcome these shortcomings Moore and
Saffman (1975) developed an asymptotic method in which the velocity of the
flow inside the vortical core is found using a force balance method, which in
turn is combined with the velocity of the outer flow, given by the
desingularized Biot-Savart law. We note, however, that these authors pre-
assumed a spatial structure of vorticity in the vortex core which is not
consistent with the Euler and Navier Stokes equations. This was noted by
Callegari and Ting (1978), who proposed a more systematic approach based on a
matched asymptotic expansion, in which the velocity field in the core is
derived from the Navier-Stokes equations. In continuation of this, Klein and
Knio (1995) succeeded in reproducing the results of Callegari and Ting (1978),
by deriving the core velocity from the vorticity equation and Biot-Savart law
directly. It should be noted that in all of the mentioned approaches, the
fluid is assumed to be incompressible and gravitational forces are excluded.
For tornadoes the Mach number can generally be assumed to be small (Varaksin
(2017); Lewellen (1993)), opening up the possibility of using different
approximation methods for low Mach number flows, such as the Boussinesq,
anelastic and pseudo-incompressible approximations, to simplify the problem.
See Ogura and Phillips (1962) and Durran (1989) for a description of the
anelastic and pseudo-incompressible approximations, respectively.
Since tornadoes can have a height of several kilometers, gravity is expected
to have a considerable influence on the motion within the vortical core.
Unfortunately, there are a limited amount of theoretical studies on buoyant
vorticies. To the best of our knowledge the only attempts at explicitly
including the force of gravity in the equations of motion was presented in
Chang and Smith (2018) and more recently in Harikrishnan  _et al._ (2023).
Chang and Smith (2018) used the force balance method of Moore and Saffmann to
demonstrate certain properties of bouyant vortex rings, which had already been
experimentally confirmed by Turner (1957). In contrast, Harikrishnan  _et al._
(2023) extended the asymptotic approach of Callegari and Ting (1978) to
include the force of gravity, but only for very weak gravitational forces; in
the regime studied by Harikrishnan  _et al._ (2023) gravity only appears at
second order in the asymptotic expansion of the Navier Stokes equation and
consequently has very little impact on the self-induced motion of the vortical
core. Harikrishnan  _et al._ (2023) showed that for closed filaments as well
as for filaments which are symmetric in some streamwise direction, the effect
of gravity disappears entirely when gravity only enters at second order. We
wish to build on the work done by Harikrishnan  _et al._ (2023) and further
extend the analysis of Callegari and Ting (1978) to include the force of
gravity at first order. This corresponds to a regime in which gravity makes a
more significant contribution to the self induced motion of the filament, and
is equivalent to the regime discussed by Chang and Smith (2018). We will
refrain from doing the fully compressible analysis, as was done by Ting,
Klein, and Knio (2007), but rather focus on the Boussinesq regime as was also
done by Harikrishnan  _et al._ (2023), in which the density variations only
enter in the gravity term. We also neglect compressibility effects in the
background density and pressure stratifications in this first approach to the
problem. We emphasize that for vorticies with significant vertical span, such
as tornadoes, the Boussinesq approximation will not hold, and one should
instead apply the anelastic or pseudo-incompressible approximations to
accurately capture the change in pressure with increasing height. However, we
still consider this the appropriate starting point, as already at this level
of complexity new and interesting phenomena are expected to make an
appearance.
Our primary objective is thus to reconstruct the results of Chang and Smith
(2018) using a different method for estimating the flow inside the vortical
core. We will restrict our analysis to finding the leading order filament
velocity, and refer to Fukumoto and Miyazaki (1991) and Margerit and Brancher
(1997) for how to obtain the first order analysis. Our secondary objective is
to shed some further light on the effect of gravity on the motion of slender
vortex filaments, in particular as it relates to vorticies of a more
destructive nature, noting that further analysis with different approximation
methods will be necessary for an accurate description of vorticies of this
type.

##  II Outer Solution

The vorticity field $\bm{\Omega}(\bm{x},t)$ of the flow to be considered here
can be composed additively from the vorticity of the background flow and that
of the concentrated vortex itself. By the Biot-Savart law, the velocity
$\bm{v}(\bm{x},t)$ at any point is then composed of a velocity
$\mathbf{Q}_{1}$ due to the self-induced motion of the vortex filament, and a
velocity $\mathbf{Q}_{2}$ given by the background flow field. The velocity
$\mathbf{Q}_{1}$ at any point $\bm{x}$ not too close to the vortex core, is
given by the line Biot-Savart law for an infinitely concentrated vortex:

$\mathbf{Q}_{1}(\bm{x},t)=-\frac{\Gamma}{4\pi}\int_{\mathcal{L}}\frac{\left(\bm{x}-\mathbf{X}(s^{\prime},t)\right)\times\hat{\tau}(s^{\prime},t)\>ds^{\prime}}{\left|\bm{x}-\mathbf{X}(s^{\prime},t)\right|^{3}}$
(1)

where $\mathcal{L}$ is the vortex centerline, $\mathbf{X}(s,t)$ the position
vector for any point on $\mathcal{L}$, which is parametrized by the arc length
parameter $s$. Here $\bm{x}$ denotes a point outside of the filament core,
$\Gamma$ the total, constant circulation of the filament, and $\hat{\tau}$ the
unit tangent vector.
For the problem at hand a curvilinear coordinate system is the natural choice.
The position vector of any point $\bm{x}$ in curvilinear coordinates, can be
associated to $\mathbf{X}(s,t)$ by the equation

$\bm{x}(x,y,z,t)=\mathbf{X}(s,t)+r\hat{\mathbf{r}}(\varphi,s,t)$ (2)

Here $r$ denotes the distance from the point $\bm{x}$ to the filament
$\mathbf{X}$, and $\hat{\mathbf{r}}$ is the radial unit vector for curvilinear
coordinates. See Figure 1, adapted from Callegari and Ting (1978), for an
illustration of the coordinate system. Note that in this frame, we define a
relative velocity $\mathbf{V}$ as $\bm{v}=\dot{\mathbf{X}}(s,t)+\mathbf{V}$.

A careful expansion of the integrand in the Biot-Savart formula leads to an
expression for the behaviour of $\mathbf{Q}_{1}$ as the radial distance to the
filament goes to zero:

$\displaystyle\mathbf{Q}_{1}(r\longrightarrow
0,\varphi,s,t)=\frac{\Gamma}{2\pi
r}\hat{\bm{\theta}}+\frac{\Gamma\kappa(s,t)}{4\pi}\left[\ln\left(\frac{1}{r}\right)\right]\hat{\mathbf{b}}\hskip
28.45274pt$ (3)
$\displaystyle+\frac{\Gamma\kappa(s,t)}{4\pi}(\cos\varphi)\hat{\bm{\theta}}+\mathbf{Q}_{f}+\mathcal{O}\left(r\ln
r\right)$

where $\kappa(s,t)$ is the local curvature, $\hat{\bm{\theta}}$ the unit
circumferential vector and $\hat{\mathbf{b}}$ the unit binormal vector,
associated with the point $s$ on the vortex filament $\mathbf{X}(s,t)$. The
vector $\mathbf{Q}_{f}$ is the part of the Biot-Savart integral which has a
limit as $r\longrightarrow 0$. For the details on the expansion, including the
exact representation of $\mathbf{Q}_{f}$, we refer the reader to Ting, Klein,
and Knio (2007) (see in particular sections 3.1.2 and 3.3.1). Equation (3)
cannot yield the velocity on the vortex filament itself as the expression on
the right diverges as $r\longrightarrow 0$. It means that the replacement of
the slender vortex by its infinitely concentrated vortex outer limit results
in a singular perturbation problem: that is to say, to get a finite velocity
for the slender vortex center-line, we need to account for the small but
finite thickness of the slender vortex. To get this finite velocity, we will
employ a matched asymptotic expansion, closely following Callegari and Ting
(1978), where the outer solution limit near the center-line, as given in (3),
is to be matched with the far-field limit of an inner solution derived from
the Navier-Stokes equations with suitable boundary conditions on the
centerline. The inner solution takes into account the finite thickness and the
associated vorticity core structure which was neglected in the outer solution.
We will include the force of gravity, but employ the Boussinesq approximation
so that the density variations only enter in the gravitational term.

Figure 1: A sketch of the coordinate system.
Adapted from Callegari and Ting (1978)

##  III Inner Solution

In this section, we start by presenting the equations of motion for a
Boussinesq fluid in filament attached (curvilinear) coordinates. This is then
followed by the derivation of an equation of motion for the filament through
the method of matched asymptotic expansion. The initial steps of the
derivation are the same as in Callegari and Ting (1978) and Harikrishnan  _et
al._ (2023) and are included for the sake of completeness.

###  III.1 The Inner Equations

The Navier-Stokes equation in curvilinear coordinates is

$\displaystyle\ddot{\mathbf{X}}+\left(\frac{w}{h_{3}}-\frac{r}{h_{3}}\hat{\mathbf{r}}_{t}\cdot\hat{\tau}\right)\dot{\mathbf{X}}_{s}+\frac{d\mathbf{V}}{dt}=-\frac{\mathbf{\nabla}P}{\rho}\hskip
85.35826pt$ (4) $\displaystyle+\frac{1}{\mathbf{Re}}\left(\frac{1}{\rho
h_{3}}\left(\frac{1}{h_{3}}\dot{\mathbf{X}}_{s}\right)_{s}+\frac{1}{\rho}\Delta\mathbf{V}\right)+\frac{1}{\mathbf{Fr}^{2}}\hat{\mathbf{g}}\quad\quad\quad$

where $w$ is the tangential component of the relative velocity vector
$\mathbf{V}$, and $\mathbf{Re}$ and $\mathbf{Fr}$ respectively denote the
Reynolds number and Froude number. The material derivative of $\mathbf{V}$ in
curvilinear coordinates is given by

$\frac{d\mathbf{V}}{dt}=\frac{\partial\mathbf{V}}{\partial
t}+\left(\mathbf{V}-r\frac{\partial\hat{\mathbf{r}}}{\partial
t}\right)\cdot\mathbf{\nabla}\mathbf{V}$ (5)

while the tangential stretching parameter, $h_{3}$, is

$h_{3}=\sigma[1-\kappa r\cos\varphi]=\sigma[1-\kappa
r\cos(\theta+\theta_{0})]$ (6)

where $\sigma=|\mathbf{X}_{s}|$. Here $\varphi=\theta+\theta_{0}(s,t)$ gives
the angle between the normal and radial unit vectors $\mathbf{{\hat{n}}}$ and
$\mathbf{\hat{r}}$, where $\theta_{0}$ is included to account for the torsion
$\mathcal{T}_{\text{o}}$ and thus makes the coordinate system orthogonal. It
is easier to first derive the form of the differential operators in the
$(r,\theta,s,t)$ coordinates, where the variables are orthogonal, and then
obtain the form in $(r,\varphi,s,t)$ coordinates through the following
transformation

$\left(\frac{\partial}{\partial
s}\right)_{\theta}=\left(\frac{\partial}{\partial
s}\right)_{\varphi}-\sigma\mathcal{T}_{\text{o}}\frac{\partial}{\partial\varphi}.$
(7)

In regards to the asymptotic expansion, it is simpler to use the
$(r,\varphi,s,t)$ coordinates, as it avoids having to expand expressions of
the form $\sin\varphi$ and $\cos{\varphi}$ where it is difficult to determine
the higher order terms. A complete description of the coordinate system is
given in Callegari and Ting (1978) (see in particular Appendix A and B) while
the expansion in terms of the $(r,\varphi,s,t)$ coordinates is discussed in
Margerit (2002),(Margerit, 1997). In the Boussinesq approximation the density
variation only enters in the buoyancy term, $\rho\hat{\mathbf{g}}$, and can be
neglected in the rest of the equation. This yields

$\displaystyle\left(\ddot{\mathbf{X}}+\left(\frac{w}{h_{3}}-\frac{r}{h_{3}}\hat{\mathbf{r}}_{t}\cdot\hat{\tau}\right)\dot{\mathbf{X}}_{s}+\frac{d\mathbf{V}}{dt}\right)=-\mathbf{\nabla}P\hskip
28.45274pt$ (8)
$\displaystyle+\frac{1}{\mathbf{Re}}\left(\frac{1}{h_{3}}\left(\frac{1}{h_{3}}\dot{\mathbf{X}}_{s}\right)_{s}+\Delta\mathbf{V}\right)+\rho\frac{1}{\mathbf{Fr}^{2}}\hat{\mathbf{g}}$

where the temperature and pressure dependent density, $\rho$, has been
replaced by a constant density, except in the buoyancy term.
At this stage we note that for the case of a tornado-type vortex one would, at
the very least, need to allow the background density to depend on the vertical
coordinate $z$, in accordance with the anelastic approximation. This would,
however, introduce further complications, and thus we leave this analysis for
later work.

The bouyancy term can be written as

$1+\rho^{\prime}$ (9)

where $\rho^{\prime}=\rho-1$ represents the density variation with respect to
a reference density, and is assumed to be exclusively temperature and not
pressure dependent.
This can be further rewritten by noting that the variation in density will be
exclusively due to temperature variations, and not pressure variations,
yielding

$\rho-1=-\beta T_{0}(T-1)$ (10)

where $T_{0}$ denotes the reference temperature and $\beta$ is the coefficient
of thermal expansion, which in the case of an ideal gas is equal to $1/T_{0}$.

We will analyze the system in the regime

$\left(\nu/\Gamma\right)^{1/2}=\mathbf{Re}^{-1/2}=\bar{\nu}^{1/2}\epsilon$
(11)

where $\epsilon$ is a small, dimensionless parameter and
$\bar{\nu}=\mathcal{O}(1)$ as $\epsilon\longrightarrow 0$. Additionally, we
make the assumption that $\mathbf{Fr}=\bar{\lambda}\epsilon$, where
$\bar{\lambda}=\mathcal{O}(1)$. Given this distinguished limit, (8) can be
written as

$\displaystyle\left(\ddot{\mathbf{X}}+\left(\frac{w}{h_{3}}-\frac{r}{h_{3}}\hat{\mathbf{r}}_{t}\cdot\hat{\tau}\right)\dot{\mathbf{X}}_{s}+\frac{d\mathbf{V}}{dt}\right)=-\mathbf{\nabla}P\hskip
42.67912pt$ (12)
$\displaystyle+\bar{\nu}\epsilon^{2}\left(\frac{1}{h_{3}}\left(\frac{1}{h_{3}}\dot{\mathbf{X}}_{s}\right)_{s}+\Delta\mathbf{V}\right)-\frac{\alpha(T-1)}{\epsilon^{2}}\hat{\mathbf{z}}$

where we have defined

$\alpha=-\beta T_{0}\quad\text{ and
}\quad\hat{\mathbf{z}}=-\hat{\mathbf{g}}/\bar{\lambda}^{2}$ (13)

and integrated the constant buoyancy term $-\hat{z}/\epsilon^{2}$ in $-\nabla
P$, as it derives from a potential. In the Boussinesq approximation, the
continuity equation simplifies to it’s in-compressible form, which in
curvilinear coordinates $(r,\theta,s,t)$ is given by

$\left[\left(rh_{3}u\right)_{r}+\left(h_{3}v\right)_{\theta}+r\left(w_{s}+\dot{\mathbf{X}}_{s}\cdot\hat{\tau}\right)\right]=0$
(14)

and which in $(r,\varphi,s,t)$ coordinates becomes

$\left[\left(rh_{3}u\right)_{r}+\left(h_{3}v\right)_{\varphi}+r\left(w_{s}-\sigma\mathcal{T}_{\text{o}}w_{\varphi}+\dot{\mathbf{X}}_{s}\cdot\hat{\tau}\right)\right]=0$
(15)

where we note that, (15) corresponds to equation (2) in Margerit (2002).

Since we have introduced a density variation to the flow, we also need an
energy equation to close the system. The energy equation for a Boussinesq
fluid in curvilinear coordinates, given in terms of the temperature, $T$, is

$\frac{\partial T}{\partial
t}+\left(\mathbf{V}-r\frac{\partial\hat{\mathbf{r}}}{\partial
t}\right)\cdot\nabla T=\epsilon^{2}\bar{\eta}\Delta T$ (16)

where $\bar{\eta}=\bar{\nu}/{\bar{\mu}}$ and $\bar{\mu}=\mathcal{O}(1)$ is the
Prandtl number.

Next, we introduce stretched radial coordinates

$\bar{r}=r/\epsilon$ (17)

and expand the dynamical variables in terms of $\epsilon$ as follows:

$\displaystyle
u(\bar{r},\varphi,s,t;\epsilon)=u^{(1)}(\bar{r},\varphi,s,t)+\epsilon
u^{(2)}(\bar{r},\varphi,s,t)+\cdots$ (18) $\displaystyle
v(\bar{r},\varphi,s,t;\epsilon)=\epsilon^{-1}v^{(0)}(\bar{r},\varphi,s,t)+v^{(1)}(\bar{r},\varphi,s,t)+\cdots$
(19) $\displaystyle
w(\bar{r},\varphi,s,t;\epsilon)=\epsilon^{-1}w^{(0)}(\bar{r},\varphi,s,t)+w^{(1)}(\bar{r},\varphi,s,t)+\cdots\qquad$
(20)
$\displaystyle\mathbf{X}(s,t;\epsilon)=\mathbf{X}^{(0)}(s,t)+\epsilon\mathbf{X}^{(1)}(s,t)+\cdots\qquad$
(21)

Here $u,v$ and $w$ denote the radial, circumferential and tangential
components of the velocity vector $\mathbf{V}$, respectively. Furthermore, to
obtain non-trivial velocities $v^{(0)}$ and $w^{(0)}$, we must have that

$P(\bar{r},\varphi,s,t;\epsilon)=\epsilon^{-2}P^{(0)}(\bar{r},\varphi,s,t)+\epsilon^{-1}P^{(1)}(\bar{r},\varphi,s,t)+\cdots\\\
$ (22)

The perturbation temperature $\tilde{T}=T-1$ is expanded in an asymptotic
series as follows

$\tilde{T}(\bar{r},\varphi,s,t;\epsilon)=\tilde{T}^{(0)}(\bar{r},\varphi,s,t)+\epsilon\tilde{T}^{(1)}(\bar{r},\varphi,s,t)+\cdots$
(23)

In addition, we require that

$\dot{\mathbf{X}}\cdot\hat{\tau}=0$ (24)

so that the filament centerline $\mathcal{L}$ forms a material curve.
The geometric parameters $\sigma$, $\kappa$ and $h_{3}$ are, through the
Serret-Frenet formulas, functions of $\mathbf{X}(s,t)$, and thus expanded as
follows:

$\displaystyle\sigma(s,t;\epsilon)$ $\displaystyle=$
$\displaystyle\sigma^{(0)}(s,t)+\epsilon\sigma^{(1)}(s,t)+\cdots$
$\displaystyle=$
$\displaystyle|\mathbf{X}^{(0)}_{s}|+\epsilon\frac{\mathbf{X}^{(0)}_{s}\cdot\mathbf{X}^{(1)}_{s}}{|\mathbf{X}^{(0)}_{s}|}+\cdots$
$\displaystyle\kappa(s,t;\epsilon)$ $\displaystyle=$
$\displaystyle\kappa^{(0)}(s,t)+\epsilon\kappa^{(1)}(s,t)+\cdots$ (26)
$\displaystyle\mathcal{T}_{\text{o}}(s,t;\epsilon)$ $\displaystyle=$
$\displaystyle\mathcal{T}_{\text{o}}^{(0)}(s,t)+\epsilon\mathcal{T}_{\text{o}}^{(1)}(s,t)+\cdots$
(27) $\displaystyle h_{3}(\bar{r},\varphi,s,t;\epsilon)$ $\displaystyle=$
$\displaystyle\sigma^{(0)}(s,t)+\epsilon
h_{3}^{(1)}(\bar{r},\varphi,s,t;\epsilon)+\cdots$ $\displaystyle=$
$\displaystyle\sigma^{(0)}+\epsilon\left[\sigma^{(1)}-\sigma^{(0)}\kappa^{(0)}\bar{r}\cos\varphi\right]+\cdots$

Let $z_{r}$, $z_{\theta}$ and $z_{\text{t}}$ denote the radial,
circumferential and tangential components of the constant vector
$\hat{\mathbf{z}}$, such that
$\hat{\mathbf{z}}=z_{r}\hat{\mathbf{r}}+z_{\theta}\hat{\bm{\theta}}+z_{\text{t}}\hat{\bm{\tau}}$.
Equivalently, we can define
$\hat{\mathbf{z}}=z_{\text{t}}\hat{\bm{\tau}}+z_{\text{n}}\hat{\mathbf{n}}+z_{\text{b}}\hat{\mathbf{b}}$.
The expansion of the components of $\mathbf{\hat{z}}$ is thus given as:

$\displaystyle z_{r}(\varphi,s,t;\epsilon)=z_{r}^{(0)}(\varphi,s,t)+\epsilon
z_{r}^{(1)}(\varphi,s,t)+\cdots$ $\displaystyle
z_{\theta}(\varphi,s,t;\epsilon)=z_{\theta}^{(0)}(\varphi,s,t)+\epsilon
z_{\theta}^{(1)}(\varphi,s,t)+\cdots$ $\displaystyle
z_{\text{t}}(s,t;\epsilon)=z_{\text{t}}^{(0)}(s,t)+\epsilon
z_{\text{t}}^{(1)}(s,t)+\cdots$ $\displaystyle
z_{\text{n}}(s,t;\epsilon)=z_{\text{n}}^{(0)}(s,t)+\epsilon
z_{\text{n}}^{(1)}(s,t)+\cdots$ $\displaystyle
z_{\text{b}}(s,t;\epsilon)=z_{\text{b}}^{(0)}(s,t)+\epsilon
z_{\text{b}}^{(1)}(s,t)+\cdots$

Given our choice of $\epsilon$, the viscosity terms will only enter at second
order. One can show that the term

$\left(\frac{1}{h_{3}}\left(\frac{1}{h_{3}}\dot{\mathbf{X}}_{s}\right)_{s}\right)$
(29)

is $\mathcal{O}(1)$, so this term only enters much later in the expansion.

Given this, the leading order momentum equations in the Boussinesq regime are

$\displaystyle\frac{w_{\varphi}^{(0)}v^{(0)}}{\bar{r}}$ $\displaystyle=$
$\displaystyle 0$ (30a)
$\displaystyle\frac{v^{(0)}v_{\varphi}^{(0)}}{\bar{r}}$ $\displaystyle=$
$\displaystyle-\frac{1}{\bar{r}}P_{\varphi}^{(0)}$ (30b)
$\displaystyle\frac{\left(v^{(0)}\right)^{2}}{\bar{r}}$ $\displaystyle=$
$\displaystyle P_{\bar{r}}^{(0)}$ (30c)

while the leading order continuity and temperature equations are

$v_{\varphi}^{(0)}=0$ (31)

and

$T_{\varphi}^{(0)}=0$ (32)

Combining (30a), (30b), (31) and (32) we conclude that all the leading order
field components are independent of $\varphi$.
Thus, dropping all the terms containing leading order $\varphi$-derivatives,
the first order momentum equations are

$\displaystyle\Bigg{(}u^{(1)}w_{\bar{r}}^{(0)}+\frac{w_{\varphi}^{(1)}}{\bar{r}}v^{(0)}+\frac{w^{(0)}w^{(0)}_{s}}{\sigma^{(0)}}\hskip
71.13188pt$ (33a)
$\displaystyle+w^{(0)}\kappa^{(0)}v^{(0)}\sin\varphi\Bigg{)}=-\frac{P_{s}^{(0)}}{\sigma^{(0)}}-\alpha\tilde{T}^{(0)}z_{\text{t}}^{(0)}$
$\displaystyle\Bigg{(}u^{(1)}v_{\bar{r}}^{(0)}+\frac{v^{(0)}v_{\varphi}^{(1)}}{\bar{r}}+\frac{v^{(0)}u^{(1)}}{\bar{r}}+\frac{w^{(0)}v^{(0)}_{s}}{\sigma^{(0)}}\hskip
42.67912pt$ (33b)
$\displaystyle-\left(w^{(0)}\right)^{2}\kappa^{(0)}\sin\varphi\Bigg{)}=-\frac{P_{\varphi}^{(1)}}{\bar{r}}-\alpha\tilde{T}^{(0)}z_{\theta}^{(0)}$
$\displaystyle\left(\frac{v^{(0)}}{\bar{r}}u^{(1)}_{\varphi}-\frac{2v^{(0)}v^{(1)}}{\bar{r}}+\left(w^{(0)}\right)^{2}\kappa^{(0)}\cos\varphi\right)\hskip
42.67912pt$ (33c)
$\displaystyle=-P_{\bar{r}}^{(1)}-\alpha\tilde{T}^{(0)}z_{r}^{(0)}$

where

$\displaystyle z_{r}^{(0)}$ $\displaystyle=$ $\displaystyle
z_{n}^{(0)}\cos\varphi+z_{b}^{(0)}\sin\varphi$ (34) $\displaystyle
z_{\theta}^{(0)}$ $\displaystyle=$ $\displaystyle z_{b}^{(0)}\cos\varphi-
z_{n}^{(0)}\sin\varphi$ (35)

For the first order continuity equation we get

$\displaystyle\sigma^{(0)}\left(v_{\varphi}^{(1)}+\left(\bar{r}u^{(1)}\right)_{\bar{r}}\right)+\bar{r}w_{s}^{(0)}\hskip
56.9055pt$ (36)
$\displaystyle+\bar{r}\kappa^{(0)}\sigma^{(0)}\sin\varphi\;v^{(0)}=0$

while the first order temperature equations is

$\frac{v^{(0)}}{\bar{r}}T_{\varphi}^{(1)}+u^{(1)}T_{\bar{r}}^{(0)}+\frac{w^{(0)}}{\sigma^{(0)}}T_{s}^{(0)}=0$
(37)

Note that the right hand side of (16) is $\mathcal{O}(1)$ and thus has no
impact on the first two order temperature equations. Similarly, the torsion
term appearing in (15) is $\mathcal{O}(\epsilon)$ and thus only enters in the
2nd order equations.

For the inner solution, we additionally have

$\displaystyle v^{(i)}=0,\quad u^{(i)}=0,\quad\quad i=0,1,2,...\quad\text{ at
}\bar{r}=0$ (38)

which comes directly from the definition of the relative velocity
$\mathbf{V}$.

###  III.2 The evolution equations for the filament

To leading order, the outer flow equation is axially symmetric (i.e.
independent of $s$). Consequently, it would make sense to make a similar
requirement for the inner solution, as was indeed done by Callegari and Ting
(1978) and Harikrishnan  _et al._ (2023). In particular this would mean making
the assumption that $v_{s}^{(0)}=0$. However, by averaging (33a) with respect
to $\varphi$ we get

$u_{c}^{(1)}w^{(0)}_{\bar{r}}+\frac{w^{(0)}w^{(0)}_{s}}{\sigma^{(0)}}=-\frac{P_{s}^{(0)}}{\sigma^{(0)}}-\alpha\tilde{T}^{(0)}z_{\text{t}}^{(0)}$
(39)

where

$f_{c}=\frac{1}{2\pi}\int_{0}^{2\pi}f(\varphi)\;d\varphi$ (40)

denotes the average with respect to $\varphi$, or equivalently the symmetric
part, of the function $f$. Applying the same averaging procedure to (33b) and
(36) yields

$\displaystyle\frac{1}{\bar{r}}\left(\bar{r}v^{(0)}\right)_{\bar{r}}\;u_{c}^{(1)}+\frac{w^{(0)}v_{s}^{(0)}}{\sigma^{(0)}}=0$
(41)
$\displaystyle\sigma^{(0)}\left(\bar{r}u_{c}^{(1)}\right)_{\bar{r}}+\bar{r}w_{s}^{(0)}=0$
(42)

Hence, if $v_{s}^{(0)}=0$, we must have that

$u^{(1)}_{c}=0\quad\text{ and }\quad w_{s}=0$ (43)

but we cannot from this conclude that $P_{s}^{(0)}=0$, due to the buoyancy
term appearing in (39). Unless the buoyancy term vanishes, this leads to a
contradiction as 30c) relates $P^{(0)}$ directly to $v^{(0)}$. Thus, we
necessarily must require that the leading order components are all axially
dependent. The case of vortex filaments with leading order axial dependence
was treated by Klein and Ting (1992), again excluding the force of gravity.

Since the first order equations are linear with respect to the first order
field components, $u^{(1)},v^{(1)},w^{(1)},\tilde{T}^{(1)}$ and $P^{(1)}$, we
can decompose the solution into symmetric and asymmetric parts, i.e.

$f(\varphi)=f_{c}+f_{a}(\varphi)$

with $f_{c}$ given by (40). It is interesting to to note that when splitting
(33) between its symmetric and asymmetric parts, the added buoyancy term in
(33a) will impact only the symmetric part of the equations. On the contrary,
the added buoyancy term in (33b) and (33c) will impact only the asymmetric
part of the equations. The asymmetric first order momentum equations are thus:

$\displaystyle
u_{a}^{(1)}w_{\bar{r}}^{(0)}+v^{(0)}\frac{\left(w_{a}^{(1)}\right)_{\varphi}}{\bar{r}}+w^{(0)}\kappa^{(0)}v^{(0)}\sin\varphi=0\qquad\qquad$
(44a) $\displaystyle
u_{a}^{(1)}v_{\bar{r}}^{(0)}+\frac{v^{(0)}\left(v^{(1)}_{a}\right)_{\varphi}}{\bar{r}}+\frac{v^{(0)}u^{(1)}_{a}}{\bar{r}}-\left(w^{(0)}\right)^{2}\kappa^{(0)}\sin\varphi$
(44b)
$\displaystyle=-\frac{\left(P^{(1)}_{a}\right)_{\varphi}}{\bar{r}}-\alpha\tilde{T}^{(0)}z_{\theta}^{(0)}\qquad\qquad$
$\displaystyle\frac{v^{(0)}}{\bar{r}}\left(u_{a}^{(1)}\right)_{\varphi}-\frac{2v^{(0)}v_{a}^{(1)}}{\bar{r}}+\left(w^{(0)}\right)^{2}\kappa^{(0)}\cos\varphi$
(44c)
$\displaystyle=-\left(P^{(1)}_{a}\right)_{\bar{r}}-\alpha\tilde{T}^{(0)}z_{r}^{(0)}\qquad\qquad$

Hence, we conclude that since the $s$-derivatives appear exclusively in the
symmetric terms, these terms do not appear in the asymmetric first order
equations, from which we will derive the filament equation of motion and the
associated core constant. As a final remark on the topic of the gravity
induced axial asymmetry, we note that if the vortex filament is placed
horizontally such that the term proportional to $z_{\text{t}}^{(0)}$ vanishes,
one would again be permitted to assume axial independence of the leading order
velocity components. Thus, it is only for filaments that are tilted with
respect to the vertical for which the axial dependence necessarily enters.

We continue by first noting that (44a) decouples from the other first order
momentum equations, by which we mean that first order tangential velocity
component, $w^{(1)}$ only appears in this equation and not in the other two.
Furthermore, we can eliminate $P_{c}^{(1)}$ from (44b) and (44c) through cross
differentiation. Through introduction of the asymmetric stream function
$\psi^{(1)}$

$u_{a}^{(1)}=\frac{1}{\bar{r}}\psi_{\varphi}^{(1)}\quad,\quad
v_{a}^{(1)}=-\psi^{(1)}_{\bar{r}}+\bar{r}\kappa^{(0)}v^{(0)}\cos\varphi$ (45)

and the leading order axial vorticity, $\zeta^{(0)}$

$\zeta^{(0)}=\frac{1}{\bar{r}}\left(\bar{r}v^{(0)}\right)_{\bar{r}}$ (46)

the two first order momentum equations, (44b) and (44c), are transformed into
one differential equation for the stream function $\psi^{(1)}$:

$\displaystyle
v^{(0)}\Delta\psi^{(1)}_{\varphi}-\zeta_{\bar{r}}^{(0)}\psi^{(1)}_{\varphi}=$
$\displaystyle-$ $\displaystyle\kappa^{(0)}\sin\varphi\;H(\bar{r},s,t)$ (47)
$\displaystyle+$ $\displaystyle\alpha B(\bar{r},s,t)z_{\theta}^{(0)}$

where $\Delta$ denotes the Laplacian of $\bar{r}$ and $\varphi$ and

$\displaystyle H(\bar{r},s,t)$ $\displaystyle=$ $\displaystyle
2\bar{r}\zeta^{(0)}v^{(0)}+\left(v^{(0)}\right)^{2}+2\bar{r}w^{(0)}w^{(0)}_{\bar{r}}$
(48) $\displaystyle B(\bar{r},s,t)$ $\displaystyle=$
$\displaystyle\bar{r}\tilde{T}^{(0)}_{\bar{r}}$ (49)

We expand $\psi^{(1)}$ in a Fourier series as follows

$\psi^{(1)}=\sum_{n=1}^{\infty}\left(\psi_{n1}^{(1)}\cos
n\varphi+\psi_{n2}^{(1)}\sin n\varphi\right)$ (50)

By using (50) in (47) and identifying all Fourier modes in the equation, we
get an equation for each mode. This set of equations can be written in the
following compact form

$\displaystyle
v^{(0)}\left[\frac{1}{\bar{r}}\frac{\partial}{\partial\bar{r}}+\frac{\partial^{2}}{\partial\bar{r}^{2}}-\left(\frac{n^{2}}{\bar{r}^{2}}+\frac{\zeta_{\bar{r}}^{(0)}}{v^{(0)}}\right)\right]\psi^{(1)}_{nj}=\Upsilon^{(0)}_{nj}$
(51)

where

$\Upsilon^{(0)}_{nj}=\left[\kappa^{(0)}H+\alpha
z_{\text{n}}^{(0)}B\right]\delta_{n1}\delta_{j1}+\alpha
z_{\text{b}}^{(0)}B\delta_{n1}\delta_{j2}$ (52)

The boundary conditions for (51) are given by (38) and the farfield
conditions. The latter are found by comparing the inner expansion of the
circumferential and tangential velocity components, (19) and (20), with the
outer expansion, (3). For the leading order velocity components we get

$\displaystyle v^{(0)}$ $\displaystyle\sim$
$\displaystyle\frac{\Gamma}{2\pi\bar{r}}\quad\text{ as
}\bar{r}\longrightarrow\infty$ (53a) $\displaystyle w^{(0)}$
$\displaystyle\sim$ $\displaystyle o(\bar{r}^{-n})\quad\text{ for all }n\text{
as }\bar{r}\longrightarrow\infty$ (53b)

while for the first order components we get

$\displaystyle v^{(1)}$ $\displaystyle\longrightarrow$
$\displaystyle\frac{\Gamma\kappa^{(0)}}{4\pi}\cos\varphi\left(\ln\frac{1}{\epsilon\bar{r}}+1\right)$
(54a)
$\displaystyle+\left(\mathbf{Q}_{0}-\dot{\mathbf{X}}^{(0)}\right)\cdot\hat{\theta}^{(0)}$
$\displaystyle u^{(1)}$ $\displaystyle\longrightarrow$
$\displaystyle\frac{\Gamma\kappa^{(0)}}{4\pi}\sin\varphi\ln\frac{1}{\epsilon\bar{r}}+\left(\mathbf{Q}_{0}-\dot{\mathbf{X}}^{(0)}\right)\cdot\hat{\mathbf{r}}^{(0)}\hskip
28.45274pt$ (54b) $\displaystyle w^{(1)}$ $\displaystyle\longrightarrow$
$\displaystyle\mathbf{Q}_{0}\cdot\hat{\tau}^{(0)}$ (54c)

as $\bar{r}\longrightarrow\infty$. Here
$\mathbf{Q}_{0}=\left(\mathbf{Q}_{f}+\mathbf{Q}_{2}(r=0)\right)$.
Given this, (51) implies that

$\psi_{n1}^{(1)}=0\quad,\quad\psi_{n2}^{(1)}=0\quad\text{for }n\neq 1$

The inhomogeneous solutions to (51), for $n=1$, that satisfies the boundary
condition at $\bar{r}=0$ is given by

$\psi_{1j}^{(1)}=v^{(0)}\int_{0}^{\bar{r}}\frac{1}{z(v^{(0)})^{2}}\left[\int_{0}^{z}\xi\Upsilon^{(0)}_{nj}(\xi,s,t)\;d\xi\right]dz$
(55)

To confirm this assertion, observe that both $v^{(0)}$ and
$v^{(0)}\int_{0}^{\bar{r}}\frac{1}{z\left(v^{(0)}\right)^{2}}\;dz$ are
solutions to the homogeneous version of (51). This provides the
$\psi^{(1)}_{1j}$ functions

$\psi^{(1)}_{11}=\kappa^{(0)}v^{(0)}J+\alpha z_{\text{n}}^{(0)}v^{(0)}I$ (56a)
and $\psi^{(1)}_{12}=\alpha z_{\text{b}}^{(0)}v^{(0)}I$ (56b)

where

$J=\int_{0}^{\bar{r}}\frac{1}{z(v^{(0)})^{2}}\left[\int_{0}^{z}\xi
H(\xi,s,t)\;d\xi\right]dz$ (57a) and
$I=\int_{0}^{\bar{r}}\frac{1}{z(v^{(0)})^{2}}\left[\int_{0}^{z}{\xi}^{2}\tilde{T}^{(0)}_{\xi}\;d\xi\right]dz$
(57b)

as $B(\xi,s,t)=\xi\tilde{T}^{(0)}_{\xi}$.

It remains to determine the correct scaling for the leading order temperature
difference $\tilde{T}^{(0)}$. The temperature satisfies the parabolic-
transport equation. Thus, since $\tilde{T}^{(0)}$ is concentrated on the
vortex initially, we can conclude that the temperature should decay
exponentially as $\bar{r}$ goes to infinity for all later time $t$. In other
words,

$\tilde{T}^{(0)}\sim o(\bar{r}^{-n})\quad\text{for all }n\text{ as
}\bar{r}\longrightarrow\infty$ (58)

For completeness we note that in the more realistic case of non-zero Mach
number, the temperature also depends on the pressure, which would yield a
different scaling for the leading order temperature difference; see Ting,
Klein, and Knio (2007) for a description of fully compressible vortex flows,
excluding the force of gravity.

Inserting (53a), (53b) and (58) into (52) yields,

$\Upsilon^{(0)}_{nj}\sim\frac{\Gamma\kappa^{(0)}}{2\pi\bar{r}}$ (59)

as $\bar{r}\longrightarrow\infty$. By integrating (51) for large $\bar{r}$
twice, we get

$\displaystyle\psi_{11}^{(1)}$ $\displaystyle\sim$
$\displaystyle\frac{\kappa^{(0)}\Gamma}{4\pi}\bar{r}\ln\bar{r}+C_{1}\bar{r}$
(60a) $\displaystyle\psi_{12}^{(1)}$ $\displaystyle\sim$ $\displaystyle
C_{2}\bar{r}$ (60b)

where $C_{1}$ and $C_{2}$ are integration constants which together are
referred to as the core constant. As will be demonstrated shortly, $C_{1}$ and
$C_{2}$ correspond to the binormal and normal parts of the core constant and
couple the velocity of the filament to the motion inside the filament core.

Comparing (56a) and (60a) we find that

$\displaystyle C_{1}(s,t)$ $\displaystyle=$
$\displaystyle\lim_{\bar{r}\longrightarrow\infty}(\psi^{(1)}_{11}-\frac{\kappa^{(0)}\Gamma}{4\pi}\bar{r}\ln\bar{r})/\bar{r}$
$\displaystyle=$
$\displaystyle\frac{\Gamma}{4\pi}\left[\kappa^{(0)}\lim_{\bar{r}\longrightarrow\infty}(J-h\ln\bar{r})/h+\alpha
z_{\text{n}}^{(0)}\lim_{\bar{r}\longrightarrow\infty}I/h\right]$

where $h=\frac{\Gamma}{4\pi}\bar{r}/v^{(0)}$, and $h(\infty)=\bar{r}^{2}/2$.

$\displaystyle
C_{2}(s,t)=\lim_{\bar{r}\longrightarrow\infty}(\psi^{(1)}_{12}/\bar{r})=\frac{\Gamma}{4\pi}\alpha
z_{\text{b}}^{(0)}\lim_{\bar{r}\longrightarrow\infty}I/h$ (62)

The Hopital’s rule is applied to those limits: having $f(x)$ and $g(x)$ that
goes to infinity at $x=\infty$, if $f_{x}/g_{x}(\infty)=l$ then
$f/g(\infty)=l$. Hence, we get

$\displaystyle\lim_{\bar{r}\longrightarrow\infty}(J-h\ln\bar{r})/h=C_{v}(s,t)+C_{w}(s,t)$
(63)

where

$\displaystyle C_{v}(s,t)$ $\displaystyle=$
$\displaystyle\lim_{\bar{r}\longrightarrow\infty}\left(\frac{4\pi^{2}}{\Gamma^{2}}\int_{0}^{\bar{r}}\xi\left(v^{(0)}\right)^{2}\;d\xi-\ln\bar{r}\right)+\frac{1}{2}$
(64) $\displaystyle C_{w}(s,t)$ $\displaystyle=$
$\displaystyle-\frac{8\pi^{2}}{\Gamma^{2}}\int_{0}^{\infty}\xi\left(w^{(0)}\right)^{2}\;d\xi$
(65)

The details of this calculation are included in Appendix A. Similarly,

$\displaystyle C_{T}(s,t)=\lim_{\bar{r}\longrightarrow\infty}I/d$
$\displaystyle=$
$\displaystyle\frac{4\pi^{2}}{\Gamma^{2}}\lim_{\bar{r}\longrightarrow\infty}\int_{0}^{\bar{r}}{\xi}^{2}\tilde{T}^{(0)}_{\xi}\;d\xi$
(66) $\displaystyle=$
$\displaystyle-\frac{8\pi^{2}}{\Gamma^{2}}\int_{0}^{\infty}\xi\tilde{T}^{(0)}\;d\xi$

And so we get for the core constants $C_{1}(s,t)$ and $C_{2}(s,t)$

$\displaystyle C_{1}(s,t)$ $\displaystyle=$
$\displaystyle\frac{\Gamma}{4\pi}\left[\kappa^{(0)}(C_{v}(t)+C_{w}(t))+\alpha
z_{\text{n}}^{(0)}C_{T}(t)\right]$ (67a) $\displaystyle C_{2}(s,t)$
$\displaystyle=$ $\displaystyle\frac{\Gamma}{4\pi}\alpha
z_{\text{b}}^{(0)}C_{T}(s,t)$ (67b)

To arrive at an equation of motion for the filament, we need a matching
condition for $\psi^{(1)}$ to which (60a) and (60b) can be compared. This
condition is given by the matching conditions for the leading order velocity
field components, (54a)-(54c). We get

$\displaystyle\psi^{(1)}\sim-\frac{\Gamma\kappa^{(0)}}{4\pi}\cos\varphi\;\bar{r}\ln\left(\frac{1}{\epsilon\bar{r}}\right)\hskip
56.9055pt$ (68)
$\displaystyle+\bar{r}\left[(\mathbf{Q}_{0}-\dot{\mathbf{X}}^{(0)})\cdot\left(\hat{\mathbf{n}}^{(0)}\sin\varphi-\hat{\mathbf{b}}^{(0)}\cos\varphi\right)\right]$

Comparing (60a), (60b) and (68), using (50), yields $C_{1}(s,t)$ and
$C_{2}(s,t)$ in terms of the filament velocity, from which we write

$\dot{\mathbf{X}}^{(0)}=\mathbf{Q}_{0}^{\perp}+\left(\frac{\Gamma\kappa^{(0)}}{4\pi}\ln\frac{1}{\epsilon}+C_{1}(s,t)\right)\hat{\mathbf{b}}^{(0)}-C_{2}(s,t)\hat{\mathbf{n}}^{(0)}$
(69)

where
$\mathbf{Q}_{0}^{\perp}=\mathbf{Q}_{0}-\mathbf{Q}_{0}\cdot\hat{\mathbf{\tau}}^{(0)}\hat{\mathbf{\tau}}^{(0)}$
denotes the part of $\mathbf{Q}_{0}$ which is perpendicular to $\hat{\tau}$.
Then inserting the previously found expressions for $C_{1}(s,t)$ and
$C_{2}(s,t)$, given by (67a) and (67b), into (69) gives the equation of motion

$\displaystyle\dot{\mathbf{X}}^{(0)}=\mathbf{Q}_{0}^{\perp}$ $\displaystyle+$
$\displaystyle\frac{\Gamma\kappa^{(0)}}{4\pi}\left[\ln\frac{1}{\epsilon}+C_{v}(s,t)+C_{w}(s,t)\right]\hat{\mathbf{b}}^{(0)}$
(70) $\displaystyle-$ $\displaystyle\frac{\Gamma}{4\pi}\alpha
C_{T}(s,t)\hat{\mathbf{z}}\times\hat{\mathbf{\tau}}^{(0)}$

where we used
$\hat{\mathbf{z}}\times\hat{\mathbf{\tau}}^{(0)}=\hat{\mathbf{z}}\times(\hat{\mathbf{n}}^{(0)}\times\hat{\mathbf{b}}^{(0)})=\hat{\mathbf{n}}^{(0)}z_{\text{b}}^{(0)}-\hat{\mathbf{b}}^{(0)}z_{\text{n}}^{(0)}$.
To close the system, we need equations describing the temporal and spatial
evolution of $w^{(0)}$, $v^{(0)}$, $P^{(0)}$ and $T^{(0)}$; these equations
will be derived in the next section.

We conclude this section with a discussion of the implications of the
inclusion of buoyancy for the filament equation of motion. Firstly, we note
that the binormal part of the buoyancy term appearing in (70) is maximal for a
horizontally aligned filament, and vanishes for a filament aligned in the
vertical direction. Assuming the density of the core is smaller than the
density of the surrounding fluid, such that $\tilde{T}^{(0)}$ is negative and
$C_{T}$ positive, then this term is negative whenever the angle between the
normal and vertical unit vectors, $\hat{\mathbf{n}}$ and $\hat{\mathbf{z}}$,
is between $0$ and $\pi/2$, and positive whenever the angle is between $\pi/2$
and $\pi$. Thus, this term is responsible for the tendency of buoyant vortex
rings to align themselves in the horizontal plane, as noted by Chang and Smith
(2018) . Secondly, for a circular, horizontally aligned filament, the normal
part of the buoyancy term appearing in (70) yields, provided $\tilde{T}^{(0)}$
is negative, a negative contribution to the normal component of the filament
velocity. This, in turn will cause the ring to expand, with the rate of
expansion being determined by the density difference between the filament and
the surrounding fluid. Next, for a very long vertically aligned filament we
would expect there to be a considerable density difference between the top and
the bottom of the filament. If such a filament were to be slightly tilted in
such a way that either $z_{\text{b}}^{(0)}$ or $z_{\text{n}}^{(0)}$ are no
longer identically zero, one would see a potentially significant difference in
the normal or binormal velocity component arising from the core constant,
depending on the height above ground. In that case, the lower, and presumed
denser, part of the filament would have a smaller velocity than the parts
further up, causing the filament to tilt further. However, this effect would
be competing with the effect of the tangential leading order velocity, which
has been found to reduce the velocity in the binormal direction (Callegari and
Ting (1978)), an effect that, as we shall see in the next section, is
potentially enhanced by the presence of gravity, depending on the direction of
the axial flow. Furthermore, we note that if one were to employ the anelastic
approximation and thus properly account for the stratification of the external
flow field, it is conceivable that the difference in $\tilde{T}^{(0)}$ between
the lower and upper parts of the filament would be reduced as the background
density is no longer a constant. Finally, we would like to highlight the
similarity between the gravitational terms as they appear in (67a) and (67b),
and (2.24) in Chang and Smith (2018). The only difference is that in the
latter case the density within the core is assumed to be constant, while we
allow the density of the core to continuously approach the density of the
surrounding fluid as one approaches the outer boundary. Thus, we have
succeeded in reproducing the results of Chang and Smith (2018) using a
different approach to estimate the velocity of the vortical core. It should be
noted that we have limited our analysis to finding the leading order
contribution to the filament velocity, i.e. $\dot{\mathbf{X}}^{(0)}$, and one
could in principle continue the analysis to first order following the method
outlined in Fukumoto and Miyazaki (1991)and Margerit and Brancher (1997). This
should lead to terms corresponding to the last three terms in (2.24) in Chang
and Smith (2018). For completeness, we note that one could derive (67a) and
(67b) directly from the Biot-Savart law and vorticity equation, following the
procedure outlined in Klein and Knio (1995).

##  IV Evolution Equations for the Leading Order Velocity Components and
Temperature

The evolution equations for the dynamic variables $v^{(0)}$, $w^{(0)}$,
$P^{(0)}$ and $T^{(0)}$ are derived from the first and second order symmetric
density, momentum and temperature equations. As already mentioned, due to the
gravity term appearing (33a), we cannot assume that all of the leading order
field components are independent of the axial coordinate $s$. However, we
could, thus far, seemingly ignore this issue, since at first order the
$s$-derivative terms only appear in the symmetric terms, whereas the velocity
depends on the first order Fourier modes (see Klein and Ting (1992)). Since
the compatibility conditions, from which we get the evolution equations for
the leading order field components, are given by the symmetric equations, we
will now see the appearance of terms containing leading order axial
derivatives. For this reason we cannot derive the compatibility conditions the
way outlined in Callegari and Ting (1978), but will instead follow the
procedure outlined in Klein and Ting (1992), in which the dynamic variables
were allowed to have leading order axial variation. The key observation is
that along stream lines in the $r$-$s$-plane the leading order circulation and
leading order total head (as will be defined in Section IV.B) are conserved
quantities. It should be noted, however, that in defining the s-derivate along
a stream line the assumption is made that the leading order axial velocity
$w^{(0)}$ does not vanish at any point. A discussion of the case of vanishing
axial velocity is left to Appendix C.
In broad strokes, the derivation of the evolution equations is presented as
follows: first we provide the relevant leading and first order symmetric
temperature and momentum equations. Next, the definition of the new dynamic
variables, namely circulation and total head, are given and the kinetic energy
equation, derived by contracting (12) with $(\dot{\mathbf{X}}+\mathbf{V})$, is
introduced, before we present the coordinate transformation which defines the
s-derivative along a stream line and the associated inverse operator or
integral. Finally, by combining the first and second order symmetric equations
we derive temporal evolution equations for the dynamic variables $v^{(0)}$,
$w^{(0)}$, $P^{(0)}$ and $T^{(0)}$, with the assumption that the filament
remains symmetric in some stream-wise direction.

In preparation for this derivation, we recall that we can write the leading
order field components as follows:

$f^{(1)}=f_{c}^{(1)}+f_{11}^{(1)}\cos{\varphi}+f_{12}^{(1)}\sin\varphi$ (71)

Here $f$ stand for the variables $u,w,v$ or $T$, and $f_{11}$ and $f_{12}$ are
the Fourier coefficients, which are known functions of $\psi^{(1)}_{11}$ and
$\psi^{(1)}_{12}$. From (45) we immediately have that

$\displaystyle u_{11}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{\bar{r}}\psi_{12}^{(1)}$ (72) $\displaystyle
u_{12}^{(1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{{\bar{r}}}\psi_{11}^{(1)}$ (73) $\displaystyle
v_{11}^{(1)}$ $\displaystyle=$
$\displaystyle-\left(\psi_{11}^{(1)}\right)_{\bar{r}}+\bar{r}\kappa^{(0)}v^{(0)}$
(74) $\displaystyle v_{12}^{(1)}$ $\displaystyle=$
$\displaystyle-\left(\psi^{(1)}_{12}\right)_{\bar{r}}$ (75)

while by combining (45), (37) and (44a), we get

$\displaystyle T_{11}^{(1)}$ $\displaystyle=$
$\displaystyle-\psi_{11}^{(1)}\frac{T_{\bar{r}}^{(0)}}{v^{(0)}}$ (76)
$\displaystyle T_{12}^{(1)}$ $\displaystyle=$
$\displaystyle-\psi_{12}^{(1)}\frac{T_{\bar{r}}^{(0)}}{v^{(0)}}$ (77)
$\displaystyle w_{12}^{(1)}$ $\displaystyle=$
$\displaystyle-\frac{\psi_{12}^{(1)}w_{\bar{r}}^{(0)}}{v^{(0)}}$ (78)
$\displaystyle w_{11}^{(1)}$ $\displaystyle=$
$\displaystyle-\frac{\psi_{11}^{(1)}w_{\bar{r}}^{(0)}}{v^{(0)}}+\bar{r}w^{(0)}\kappa^{(0)}$
(79)

These relations are needed when deriving the symmetric second order momentum
and temperature equations.

###  IV.1 The first and second order symmetric momentum and temperature
equations

The symmetric first order continuity and temperature equations are

$\displaystyle\sigma^{(0)}\left(\bar{r}u_{c}^{(1)}\right)_{\bar{r}}+\bar{r}w_{s}^{(0)}$
$\displaystyle=$ $\displaystyle 0$ (80a) $\displaystyle
u_{c}^{(1)}T_{\bar{r}}^{(0)}+\frac{w^{(0)}}{\sigma^{(0)}}T_{s}^{(0)}$
$\displaystyle=$ $\displaystyle 0$ (80b)

while the symmetric first order momentum equations are given by

$\displaystyle
u_{c}^{(1)}w_{\bar{r}}^{(0)}+\frac{w^{(0)}w_{s}^{(0)}}{\sigma^{(0)}}=-\frac{P_{s}^{(0)}}{\sigma^{(0)}}-\alpha\tilde{T}^{(0)}z_{\text{t}}^{(0)}$
(81a) $\displaystyle
u_{c}^{(1)}v^{(0)}_{\bar{r}}+\frac{v^{(0)}u_{c}^{(1)}}{\bar{r}}+\frac{w^{(0)}}{\sigma^{(0)}}v_{s}^{(0)}=0$
(81b)
$\displaystyle\frac{2v^{(0)}v_{c}^{(1)}}{\bar{r}}=\left(P^{(1)}_{c}\right)_{\bar{r}}$
(81c)

The symmetric part of the second order circumferential momentum equation is:

$\displaystyle
v_{t}^{(0)}+u_{c}^{(1)}\left(v^{(1)}_{c}\right)_{\bar{r}}+\frac{v^{(0)}}{\bar{r}}u_{c}^{(2)}+u_{c}^{(2)}v^{(0)}_{\bar{r}}+\frac{v_{c}^{(1)}u_{c}^{(1)}}{\bar{r}}$
(82)
$\displaystyle+\frac{w^{(0)}}{\sigma^{(0)}}\left(v_{c}^{(1)}\right)_{s}+\frac{w_{c}^{(1)}v_{s}^{(0)}}{\sigma^{(0)}}-\frac{w^{(0)}v_{s}^{(0)}\sigma^{(1)}}{\left(\sigma^{(0)}\right)^{2}}-\kappa^{(0)}w^{(0)}w_{12}^{(1)}$
$\displaystyle+\frac{1}{2}u_{11}^{(1)}\left(v_{11}^{(1)}\right)_{\bar{r}}+\frac{1}{2}u_{12}^{(1)}\left(v_{12}^{(1)}\right)_{\bar{r}}+\frac{v_{11}^{(1)}u_{11}^{(1)}}{2\bar{r}}+\frac{v_{12}^{(1)}u_{12}^{(1)}}{2\bar{r}}$
$\displaystyle=\frac{\bar{\nu}}{\bar{r}}\left(\bar{r}v_{\bar{r}}^{(0)}\right)_{\bar{r}}-\frac{\bar{\nu}}{\bar{r}^{2}}v^{(0)}-\frac{1}{2}\alpha(\tilde{T}_{11}^{(1)}z_{b}^{(0)}-\tilde{T}_{12}^{(1)}z_{n}^{(0)})$

which upon introduction of (72)-(79), using (47) and (56a), simplifies to

$\displaystyle
v_{t}^{(0)}+u_{c}^{(1)}\left(v^{(1)}_{c}\right)_{\bar{r}}+\frac{v^{(0)}}{\bar{r}}u_{c}^{(2)}+u_{c}^{(2)}v^{(0)}_{\bar{r}}+\frac{v_{c}^{(1)}u_{c}^{(1)}}{\bar{r}}$
$\displaystyle+\frac{w^{(0)}}{\sigma^{(0)}}\left(v_{c}^{(1)}\right)_{s}+\frac{w_{c}^{(1)}v_{s}^{(0)}}{\sigma^{(0)}}-\frac{w^{(0)}v_{s}^{(0)}\sigma^{(1)}}{\left(\sigma^{(0)}\right)^{2}}$
(83)
$\displaystyle=\frac{\bar{\nu}}{\bar{r}}\left(\bar{r}v_{\bar{r}}^{(0)}\right)_{\bar{r}}-\frac{\bar{\nu}}{\bar{r}^{2}}v^{(0)}+\frac{\alpha}{2}\kappa^{(0)}z_{b}^{(0)}v^{(0)}I\zeta^{(0)}$

where $I$ is given by (57b).

Next, the symmetric part of the second order tangential momentum equation is:

$\displaystyle
w_{t}^{(0)}+u_{c}^{(2)}w^{(0)}_{\bar{r}}+u_{c}^{(1)}\left(w_{c}^{(1)}\right)_{\bar{r}}+\frac{w^{(0)}}{\sigma^{(0)}}\left(w_{c}^{(1)}\right)_{s}+\frac{w^{(1)}_{c}w_{s}^{(0)}}{\sigma^{(0)}}$
$\displaystyle-\frac{\sigma^{(1)}w^{(0)}w^{(0)}_{s}}{\left(\sigma^{(0)}\right)^{2}}+\hat{\tau}^{(0)}\cdot\dot{\mathbf{X}}_{s}^{(0)}\frac{w^{(0)}}{\sigma^{(0)}}=-\frac{1}{\sigma^{(0)}}\left(P_{c}^{(1)}\right)_{s}\quad$
$\displaystyle+\frac{\sigma^{(1)}P_{s}^{(0)}}{\left(\sigma^{(0)}\right)^{2}}+\frac{\bar{\nu}}{\bar{r}}\left(\bar{r}w_{\bar{r}}^{(0)}\right)_{\bar{r}}-\alpha\tilde{T}_{c}^{(1)}z_{\text{t}}^{(0)}-\alpha\tilde{T}^{(0)}z_{t}^{(1)}$
(84)
$\displaystyle+\frac{w^{(0)}\kappa^{(0)}}{2}\left(u_{11}^{(1)}-v_{12}^{(1)}\right)-\frac{1}{2\bar{r}}\left(v_{11}^{(1)}w_{12}^{(1)}-v_{12}^{(1)}w_{11}^{(1)}\right)$
$\displaystyle-\frac{\kappa^{(0)}v^{(0)}}{2}w_{12}^{(1)}-\frac{1}{2}\left(u_{11}^{(1)}\left(w_{11}^{(1)}\right)_{\bar{r}}+u_{12}^{(1)}\left(w_{12}^{(1)}\right)_{\bar{r}}\right)$

which, when introducing (72)- (79), using (56a), simplifies to

$\displaystyle
w_{t}^{(0)}+u_{c}^{(2)}w^{(0)}_{\bar{r}}+u_{c}^{(1)}\left(w_{c}^{(1)}\right)_{\bar{r}}+\frac{w^{(0)}}{\sigma^{(0)}}\left(w_{c}^{(1)}\right)_{s}+\frac{w^{(1)}_{c}w_{s}^{(0)}}{\sigma^{(0)}}$
$\displaystyle-\frac{\sigma^{(1)}w^{(0)}w^{(0)}_{s}}{\left(\sigma^{(0)}\right)^{2}}+\hat{\tau}^{(0)}\cdot\dot{\mathbf{X}}_{s}^{(0)}\frac{w^{(0)}}{\sigma^{(0)}}=-\frac{1}{\sigma^{(0)}}\left(P_{c}^{(1)}\right)_{s}\quad$
(85)
$\displaystyle+\frac{\bar{\nu}}{\bar{r}}\left(\bar{r}w_{\bar{r}}^{(0)}\right)_{\bar{r}}-\alpha\tilde{T}_{c}^{(1)}z_{\text{t}}^{(0)}-\alpha\tilde{T}^{(0)}z_{t}^{(1)}$
$\displaystyle+\frac{1}{2}\kappa^{(0)}\alpha
z_{b}^{(0)}v^{(0)}I\;w^{(0)}_{\bar{r}}$

Finally, the symmetric second order temperature equation is:

$\displaystyle
T_{t}^{(0)}+\frac{w^{(0)}}{\sigma^{(0)}}\left(T_{c}^{(1)}\right)_{s}+u_{c}^{(1)}\left(T_{c}^{(1)}\right)_{\bar{r}}-\frac{\sigma^{(1)}w^{(0)}}{\left(\sigma^{(0)}\right)^{2}}T_{s}^{(0)}+u_{c}^{(2)}T^{(0)}_{\bar{r}}$
$\displaystyle+\frac{w_{c}^{(1)}}{\sigma^{(0)}}T_{s}^{(0)}=\frac{\bar{\eta}}{\bar{r}}\left(\bar{r}T^{(0)}_{\bar{r}}\right)_{\bar{r}}-\frac{1}{2}\left(u_{11}^{(1)}\left(T_{11}^{(1)}\right)_{\bar{r}}+u_{12}^{(1)}\left(T_{12}^{(1)}\right)_{\bar{r}}\right)$
$\displaystyle-\frac{1}{2\bar{r}}\left(v_{11}^{(1)}T_{12}^{(1)}-v_{12}^{(1)}T_{11}^{(1)}\right)$
(86)

which simplifies to

$\displaystyle
T_{t}^{(0)}+\frac{w^{(0)}}{\sigma^{(0)}}\left(T_{c}^{(1)}\right)_{s}+u_{c}^{(1)}\left(T_{c}^{(1)}\right)_{\bar{r}}-\frac{\sigma^{(1)}w^{(0)}}{\left(\sigma^{(0)}\right)^{2}}T_{s}^{(0)}+u_{c}^{(2)}T^{(0)}_{\bar{r}}$
$\displaystyle+\frac{w_{c}^{(1)}}{\sigma^{(0)}}T_{s}^{(0)}=\frac{\bar{\eta}}{\bar{r}}\left(\bar{r}T^{(0)}_{\bar{r}}\right)_{\bar{r}}+\frac{1}{2}\kappa^{(0)}\alpha
z_{b}^{(0)}v^{(0)}IT_{\bar{r}}^{(0)}$ (87)

###  IV.2 Change of variables and coordinate transformation

The circulation around the boundary of a disk of radius $\bar{r}$ is given by

$\displaystyle\tilde{\Gamma}(\bar{r},\varphi,s,t;\epsilon)$ $\displaystyle=$
$\displaystyle 2\pi\int_{0}^{\bar{r}}\bar{r}\zeta\;d\bar{r}$ (88)
$\displaystyle=$ $\displaystyle 2\pi\mathcal{G}(\bar{r},\varphi,s,t;\epsilon)$
(89)

where we have defined $\mathcal{G}(\bar{r},\varphi,s,t;\epsilon)=\bar{r}v$. At
leading and first order we have, respectively

$\displaystyle\mathcal{G}^{(0)}=\bar{r}v^{(0)}$ (90)
$\displaystyle\mathcal{G}^{(1)}=\bar{r}v^{(1)}$ (91)

Using (46) and (81b), equation (83) can be now written in terms of
$\mathcal{G}^{(0)}$ and $\mathcal{G}^{(1)}$:

$\displaystyle\sigma^{(0)}u_{c}^{(1)}\left(\mathcal{G}^{(1)}_{c}\right)_{\bar{r}}+\left(\sigma^{(1)}u^{(1)}_{c}+\sigma^{(0)}u_{c}^{(2)}\right)\mathcal{G}_{\bar{r}}^{(0)}$
$\displaystyle+w^{(0)}\left(\mathcal{G}_{c}^{(1)}\right)_{s}+w^{(1)}_{c}\mathcal{G}_{s}^{(0)}=-\sigma^{(0)}\mathcal{G}_{t}^{(0)}$
(92)
$\displaystyle+\sigma^{(0)}\bar{\nu}\left(\mathcal{G}_{\bar{r}\bar{r}}^{(0)}-\frac{\mathcal{G}_{\bar{r}}^{(0)}}{\bar{r}}\right)+\frac{\sigma^{(0)}\kappa^{(0)}}{2\bar{r}}\alpha
z_{b}^{(0)}I\;\mathcal{G}^{(0)}\mathcal{G}_{\bar{r}}^{(0)}$

As will be shown in the section IV.C, (92) yields the first of the required
evolution equations, for the field component $v^{(0)}$.

We define the total head as

$\displaystyle{\cal{H}}=P+\left(\dot{\mathbf{X}}+\mathbf{V}\right)^{2}/2+\alpha\epsilon^{-2}\tilde{T}\tilde{\mathcal{Z}}(\bar{r},\varphi,s,t;\epsilon)$
(93)

where $\tilde{\mathcal{Z}}$ is the height of the point
$\bm{x}(x,y,z,t)=\bm{x}(r,\varphi,s,t)$ in curvilinear coordinates, defined
through

$\displaystyle(Z-Z_{0})(x,y,z)$ $\displaystyle=$ $\displaystyle z-z_{0}$
$\displaystyle=$
$\displaystyle\tilde{\mathcal{Z}}(\bar{r},\varphi,s,t;\epsilon)$
$\displaystyle=$
$\displaystyle\left[\mathbf{X}(s,t;\epsilon)+\epsilon\bar{r}\hat{\mathbf{r}}-\mathbf{X}(0,t;\epsilon)\right]\cdot\hat{\mathbf{z}}$

Here $Z-Z_{0}$ denotes the height of any point $(x,y,z)$ with respect to the
point $Z_{0}(x,y,z)=z_{0}$ in Cartesian coordinates. The symmetric part of
$\tilde{\mathcal{Z}}$ is thus

$\displaystyle\tilde{\mathcal{Z}}_{c}(s,t{;\epsilon})$ $\displaystyle=$
$\displaystyle\hat{\mathbf{z}}\cdot\left[\mathbf{X}(s,t{;\epsilon})-\mathbf{X}(0,t{;\epsilon})\right]$
(94) $\displaystyle=$
$\displaystyle{\int_{0}^{s}\sigma(s^{\prime},t{;\epsilon})z_{\text{t}}(s^{\prime},t{;\epsilon})ds^{\prime}}$
(95)

which at leading and first order gives

$\displaystyle\tilde{\mathcal{Z}}_{c}^{(0)}(s,t)$ $\displaystyle=$
$\displaystyle\hat{\mathbf{z}}\cdot\left[\mathbf{X}^{(0)}(s,t)-\mathbf{X}^{(0)}(0,t)\right]\qquad\qquad$
$\displaystyle=$
$\displaystyle\int_{0}^{s}\sigma^{(0)}(s^{\prime},t)z_{\text{t}}^{(0)}(s^{\prime},t)ds^{\prime}\qquad\qquad$
$\displaystyle\tilde{\mathcal{Z}}_{c}^{(1)}(s,t)$ $\displaystyle=$
$\displaystyle\hat{\mathbf{z}}\cdot\left[\mathbf{X}^{(1)}(s,t)-\mathbf{X}^{(1)}(0,t)\right]$
$\displaystyle=$
$\displaystyle\int_{0}^{s}\Big{(}\sigma^{(1)}(s^{\prime},t)z_{\text{t}}^{(0)}(s^{\prime},t)$
$\displaystyle\qquad\qquad+\sigma^{(0)}(s^{\prime},t)z_{t}^{(1)}(s^{\prime},t)\Big{)}ds^{\prime}$

At leading and first order, the symmetric total head is then, respectively

$\displaystyle{\cal{H}}^{(0)}$ $\displaystyle=$ $\displaystyle
p^{(0)}+\frac{{v^{(0)}}^{2}+{w^{(0)}}^{2}}{2}+\alpha{\tilde{T}}^{(0)}\tilde{\mathcal{Z}}_{c}^{(0)}$
(96) $\displaystyle{\cal{H}}^{(1)}_{c}$ $\displaystyle=$ $\displaystyle
p^{(1)}_{c}+\left[v^{(0)}v^{(1)}_{c}+w^{(0)}w^{(1)}_{c}\right]\qquad\qquad$
$\displaystyle+\alpha\left[{\tilde{T}}^{(1)}_{c}\tilde{\mathcal{Z}}_{c}^{(0)}+\tilde{T}^{(0)}_{c}\tilde{\mathcal{Z}}_{c}^{(1)}\right].$
(97)

While not explicitly written in (96) and (97), the reader should take note
that while $\mathcal{H}_{c}^{(i)},p_{c}^{(i)},v_{c}^{(i)},w_{c}^{(i)}$ and
$\tilde{T}_{c}^{(i)}$ are functions of $(\bar{r},s,t)$,
$\tilde{\mathcal{Z}}^{(i)}_{c}$ only depends on $(s,t)$.

Thus, we see that the effect of the axial component of the gravitational
force, which does not enter in the core constant derived in the previous
section, is exclusively to alter the total head, or equivalently the kinetic
energy, in the moving frame.

By contracting (12) with $\mathbf{\dot{X}+V}$ we get an equation for the
kinetic energy, which in terms of the total head can be written as:

$\displaystyle\mathcal{H}_{t}+\mathbf{v}_{R}\cdot\nabla\mathcal{H}=P_{t}-\mathbf{v}_{E}\cdot\nabla
P+\epsilon^{-2}\alpha\tilde{T}\mathcal{Z}_{t}\qquad$ (98)
$\displaystyle+\epsilon^{2}\bar{\nu}\left(\dot{\mathbf{X}}+\mathbf{V}\right)\cdot\Delta\left(\dot{\mathbf{X}}+\mathbf{V}\right)-\epsilon^{-2}\alpha\tilde{T}\mathbf{v}_{E}\cdot\nabla\mathcal{Z}+\alpha\bar{\nu}\mathcal{Z}\Delta\tilde{T}$

where we have introduced

$\displaystyle\mathbf{v}_{R}$ $\displaystyle=$
$\displaystyle\mathbf{V}-\epsilon\bar{r}\frac{\partial\hat{\mathbf{r}}}{\partial
t}$ (99) $\displaystyle\mathbf{v}_{E}$ $\displaystyle=$
$\displaystyle\dot{\mathbf{X}}+\epsilon\bar{r}\frac{\partial\hat{\mathbf{r}}}{\partial
t}$ (100)

for notational convenience. The derivation of (98) can be found in Appendix B.

The symmetric first and second order kinetic energy equations are,
respectively

$\frac{w^{(0)}}{\sigma^{(0)}}\mathcal{H}^{(0)}_{s}+u_{c}^{(1)}\mathcal{H}^{(0)}_{\bar{r}}=0$
(101)

and

$\displaystyle
u_{c}^{(2)}\mathcal{H}^{(0)}_{\bar{r}}+u_{c}^{(1)}\left(\mathcal{H}^{(1)}_{c}\right)_{\bar{r}}+\frac{w^{(0)}}{\sigma^{(0)}}\left(\mathcal{H}_{c}^{(1)}\right)_{s}$
$\displaystyle+\frac{\sigma^{(1)}}{\sigma^{(0)}}u_{c}^{(1)}\mathcal{H}_{\bar{r}}^{(0)}+\frac{w_{c}^{(1)}}{\sigma^{(0)}}\mathcal{H}_{s}^{(0)}=-\mathcal{H}^{(0)}_{t}+P^{(0)}_{t}$
$\displaystyle+\alpha
T^{(0)}\tilde{\mathcal{Z}}^{(0)}_{t}-\left[\dot{\mathbf{X}}_{s}^{(0)}\cdot\hat{\tau}^{(0)}\right]\frac{\left(w^{(0)}\right)^{2}}{\sigma^{(0)}}$
(102)
$\displaystyle+\frac{\bar{\nu}}{\bar{r}}\left(\bar{r}w^{(0)}_{\bar{r}}\right)_{\bar{r}}w^{(0)}+\frac{\bar{\nu}\mathcal{G}^{(0)}}{\bar{r}^{2}}\left(\mathcal{G}^{(0)}_{\bar{r}\bar{r}}-\frac{\mathcal{G}^{(0)}_{\bar{r}}}{\bar{r}}\right)$
$\displaystyle+\frac{\alpha\bar{\eta}}{\bar{r}}\left(\bar{r}\tilde{T}_{\bar{r}}^{(0)}\right)_{\bar{r}}\tilde{\mathcal{Z}}^{(0)}_{c}+\frac{\alpha}{2\bar{r}}\kappa^{(0)}z_{b}^{(0)}\mathcal{G}^{(0)}I\;\mathcal{H}^{(0)}_{\bar{r}}$

In section IV.C we will see that equation (102) together with (87) and (30a)
yield the required evolution equations for $w^{(0)}$, $P^{(0)}$ and $T^{(0)}$.

Through the introduction of the symmetrical stream function
$\bar{M}=\mathcal{M}(\bar{r},s,t)$, we make the following coordinate
transformation:

$\displaystyle\left(\mathcal{M},\text{id}\right):\mathbb{R}^{3}\longrightarrow\mathbb{R}^{3}\qquad\qquad\qquad$
(103) $\displaystyle(\bar{r},s,t)\mapsto(\bar{M},s,t)\qquad\qquad\qquad$ (104)
$\displaystyle
f_{c}(\bar{r},s,t)=\bar{f}_{c}(\bar{M},s,t)=f_{c}(\mathcal{M}(\bar{r},s,t),s,t)$
(105)

where $f_{c}$ denotes any of the symmetrical field components; the variables
$(\bar{M},s,t)$ are known as the von Mises variables. The inverse
transformation is given by

$\displaystyle\left(\mathcal{R},\text{id}\right):\mathbb{R}^{3}\longrightarrow\mathbb{R}^{3}$
(106) $\displaystyle(\bar{M},s,t)\mapsto(\bar{r},s,t)$ (107)

where $\bar{\mathcal{R}}=\mathcal{M}^{-1}$. Note that $\bar{\mathcal{R}}$ only
exists at regular points $\mathbf{x}(\bar{r},s,t)$ where
$\left(\frac{\partial\mathcal{M}}{\partial\bar{r}}\right)(\mathbf{x})\neq 0$.

We define

$\displaystyle M(\bar{r},\varphi,s,t;\epsilon)$ $\displaystyle=$
$\displaystyle
2\pi\int_{0}^{\bar{r}}w(\bar{r},\varphi,s,t;\epsilon)\bar{r}\;d\bar{r}$ (108)
$\displaystyle=$ $\displaystyle 2\pi\mathcal{M}(\bar{r},\varphi,s,t;\epsilon)$
(109)

as the axial mass flux through a circular disk of radius $\bar{r}$. To see
that $\mathcal{M}$ is indeed a stream function, observe that equation (80a)
relates the leading order axial mass flux, $\mathcal{M}^{(0)}$, to the
$\varphi$-averaged first order radial velocity, $u_{c}^{(1)}$, via the
relations

$\displaystyle\mathcal{M}^{(0)}(\bar{r},s,t)=\int_{0}^{\bar{r}}w^{(0)}r^{\prime}\;dr^{\prime}=\bar{M}$
(110a)
$\displaystyle\sigma^{(0)}\bar{r}u_{c}^{(1)}=-\partial_{s}\mathcal{M}^{(0)}$
(110b)

Similarly, the symmetric second order continuity equation is satisfied by the
introduction of the first order axial mass flux, $\mathcal{M}^{(1)}$:

$\displaystyle\sigma^{(0)}\bar{r}u_{c}^{(2)}+\sigma^{(1)}\bar{r}u_{c}^{(1)}+\frac{\bar{r}^{2}}{2}\left(\dot{\mathbf{X}}^{(0)}_{s}\cdot\hat{\tau}^{(0)}\right)\qquad\qquad$
(111a) $\displaystyle\qquad\qquad-\frac{\sigma^{(0)}\kappa^{(0)}}{2}\alpha
z_{b}^{(0)}\mathcal{G}^{(0)}I=-\left(\mathcal{M}_{c}^{(1)}\right)_{s}$
$\displaystyle\bar{r}w^{(1)}_{c}=\left(\mathcal{M}_{c}^{(1)}\right)_{\bar{r}}$
(111b)

We can thus define the $s$-derivative along a stream line,
$\bar{r}=R(\bar{M},s,t)$, at instant $t$ by
$D_{s}=\partial_{s}+R_{s}\partial_{\bar{r}}$ where
$R_{s}=-\mathcal{M}^{(0)}_{s}/\mathcal{M}^{(0)}_{\bar{r}}=\sigma^{(0)}u_{c}^{(1)}/w^{(0)}$,
and it is assumed that $w^{(0)}\neq 0$. We additionally define the inverse
operation $\int_{\mathcal{S}}^{[0,s]}$, as an integration operator where
$\bar{M}$ is held fixed. Consequently, the integration
$g(\bar{M},s)=\int_{\mathcal{S}}^{[0,s]}f(\bar{M},s^{\prime})\;ds^{\prime}$ is
the solution of $D_{s}(g)(\bar{M},s)=f(\bar{M},s)$.

###  IV.3 Derivation of the evolution equations

Using the symmetric first order circumferential momentum equation, and the
first order kinetic energy equation, one can show that

$D_{s}\mathcal{G}^{(0)}=0\quad\text{ and }\quad D_{s}\mathcal{H}^{(0)}=0$
(112)

Hence, given in terms of the von Mises variables, the leading order
circulation and total head are $s$-independent, and we have

$\displaystyle\mathcal{G}^{(0)}(\bar{M},s,t)$ $\displaystyle=$
$\displaystyle\mathcal{G}^{(0)}(\bar{M},t)\;=\;\mathcal{G}^{(0)}(\bar{M}(\bar{r},s,t),t)\quad$
(113) $\displaystyle\mathcal{H}^{(0)}(\bar{M},s,t)$ $\displaystyle=$
$\displaystyle\mathcal{H}^{(0)}(\bar{M},t)\;=\;\mathcal{H}^{(0)}(\bar{M}(\bar{r},s,t),t)\quad$
(114)

In addition, from the symmetric first order temperature equation we
immediately have

$D_{s}T^{(0)}=0$ (115)

and consequently

$T^{(0)}(\bar{M},s,t)=T^{(0)}(\bar{M},t)=T^{(0)}(\bar{M}(\bar{r},s,t),t)$
(116)

We still need equations for the temporal evolution of $\mathcal{G}^{(0)}$,
$\mathcal{H}^{(0)}$ and $T^{(0)}$, which are derived from the symmetric second
order equations.

Using (110), (111) and (112) we can write the left hand side of (92) as

$\displaystyle
w^{(0)}D_{s}\left[\mathcal{G}_{c}^{(1)}-\mathcal{\mathcal{G}}^{(0)}_{\mathcal{M}^{(0)}}\mathcal{M}^{(1)}_{c}\right]-\frac{\bar{r}}{2}\mathcal{G}^{(0)}_{\bar{r}}\left(\dot{\mathbf{X}}^{(0)}_{s}\cdot\hat{\tau}^{(0)}\right)$
(117)
$\displaystyle\qquad\qquad+\frac{\sigma^{(0)}\kappa^{(0)}}{2\bar{r}}\alpha
z_{b}^{(0)}\;I\;\mathcal{G}^{(0)}\mathcal{G}^{(0)}_{\bar{r}}$

Hence, we can eliminate the first order circulation, $\mathcal{G}_{c}^{(1)}$,
by integrating (92) along a stream line. In the case of an open filament there
will still be remaining boundary terms for $\mathcal{G}_{c}^{(1)}$,
$\mathcal{M}_{c}^{(1)}$ and $\mathcal{G}_{\mathcal{M}^{(0)}}^{(0)}$.
Concretely, this amounts to specifying the leading and first order velocity
field components as well as the first order axial mass flux at the boundary.
In the case of either a closed filament or an open filament which is symmetric
along the stream line (as in Klein and Ting (1992)), we get the following
evolution equation for the leading order circulation $\mathcal{G}^{(0)}$:

$\displaystyle\int_{\mathcal{S}}^{[0,s]}\Bigg{[}-\mathcal{G}_{t}^{(0)}+\frac{\bar{r}}{2\sigma^{(0)}}\mathcal{G}_{\bar{r}}^{(0)}\left(\dot{\mathbf{X}}^{(0)}_{s}\cdot\hat{\mathbf{\tau}}^{(0)}\right)\hskip
56.9055pt$ (118)
$\displaystyle+\bar{\nu}\left(\mathcal{G}^{(0)}_{\bar{r}\bar{r}}-\frac{\mathcal{G}_{\bar{r}}^{(0)}}{\bar{r}}\right)\Bigg{]}\frac{\sigma^{(0)}ds^{\prime}}{w^{(0)}}=0$

where $\mathcal{S}$ denotes the integral over a streamline.
Note that the terms containing $I\;z_{b}^{(0)}$ (or equivalently
$\psi_{12}^{(1)}$) which came from the Fourier component containing terms
appearing in equation (83) have cancelled out.

The temporal evolution equation for the leading order total head,
$\mathcal{H}^{(0)}$, is given by (102). The left hand side of (102) equation
has the same structure as the left hand side of (92), hence we can employ the
same trick and write the right hand side as:

$\displaystyle\frac{w^{(0)}}{\sigma^{(0)}}D_{s}\left[\mathcal{H}_{c}^{(1)}-\mathcal{\mathcal{H}}^{(0)}_{\mathcal{M}^{(0)}}\mathcal{M}_{c}^{(1)}\right]-\frac{\bar{r}}{2\sigma^{(0)}}\mathcal{H}^{(0)}_{\bar{r}}\left(\dot{\mathbf{X}}^{(0)}_{s}\cdot\hat{\tau}^{(0)}\right)$
$\displaystyle\qquad\qquad+\frac{\kappa^{(0)}}{2\bar{r}}\alpha
z_{b}^{(0)}\;I\;\mathcal{G}^{(0)}\mathcal{H}^{(0)}_{\bar{r}}$ (119)

Thus, by integrating along a stream line, assuming an axially symmetric
filament, we get an equation for the leading order total head,
$\mathcal{H}^{(0)}\;$ 111 Equation (120) can also be derived in a different
way without reference to the kinetic energy equation, but instead through
multiplication of each component of the momentum equation at each order by
associated components. Initially this was how we derived (120), however, upon
realisation that it may be simpler to use the kinetic equation, we then re-
derived the equation. The former derivation thus becomes a cross check of this
calculation, as the two derivations give the same result. This also explains
why we have not introduced the kinetic equation in section III.A with the
other equations: it is only when deriving equation (120) that it makes sense
to use and introduce the kinetic energy equation, which has, as far as we are
aware, never been written before in the context of slender vortex expansions.:

$\displaystyle\int_{\mathcal{S}}^{[0,s]}\Bigg{[}-\mathcal{H}^{(0)}_{t}+P^{(0)}_{t}+\alpha\tilde{T}^{(0)}\left(\tilde{\mathcal{Z}}_{c}^{(0)}\right)_{t}+\left[\frac{\bar{r}\mathcal{H}^{(0)}_{\bar{r}}}{2\sigma^{(0)}}-\frac{\left(w^{(0)}\right)^{2}}{\sigma^{(0)}}\right]\dot{\sigma}^{(0)}+\frac{\bar{\nu}}{\bar{r}}\left(\bar{r}w^{(0)}_{\bar{r}}\right)_{\bar{r}}w^{(0)}$
(120)
$\displaystyle+\frac{\bar{\nu}\mathcal{G}^{(0)}}{\bar{r}^{2}}\left(\mathcal{G}^{(0)}_{\bar{r}\bar{r}}-\frac{\mathcal{G}^{(0)}_{\bar{r}}}{\bar{r}}\right)+\frac{\alpha\bar{\eta}}{\bar{r}}\left(\bar{r}\tilde{T}_{\bar{r}}^{(0)}\tilde{\mathcal{Z}}^{(0)}_{c}\right)_{\bar{r}}\Bigg{]}\frac{\sigma^{(0)}ds^{\prime}}{w^{(0)}}=0$

Again, we note that for an open filament which is not symmetric in some
stream-wise direction, the boundary terms have to be specified. In particular,
to fix the leading order total head at the boundary, in addition to specifying
the leading and first order velocity components, the leading order pressure
and temperature components in addition to the potential differential between
the end points as determined by the integration over the streamline denoted by
$\mathcal{S}$, must be specified.

The introduction of gravitational forces leads to a modification of the
equations determining the leading order field components, in particular the
leading order tangential velocity component, $w^{(0)}$. For the incompressible
case analysed by Callegari and Ting (1978), the authors were able to conclude
that the effect of the tangential velocity is to reduce the velocity of the
filament in the binormal direction. Thus, we would expect, depending on the
sign of $\tilde{T}^{(0)}$ and $z_{\text{t}}^{(0)}$, that the presence of
gravity either enhances or reduces this effect.

To close the system we need a third compatibility condition, namely a
condition for the leading order temperature, $T^{(0)}$. Using the first and
second order symmetric temperature equations, we get can rewrite (87) as

$\displaystyle\sigma^{(0)}u_{c}^{(1)}\left(T_{c}^{(1)}\right)_{\bar{r}}+w^{(0)}\left(T_{c}^{(1)}\right)_{s}+w^{(1)}T_{s}^{(0)}+u_{c}^{(2)}T_{\bar{r}}^{(0)}$
$\displaystyle+\sigma^{(1)}u_{c}^{(1)}T_{\bar{r}}^{(0)}=-\sigma^{(0)}T_{t}^{(0)}+\frac{\sigma^{(0)}\bar{\eta}}{\bar{r}}\left(\bar{r}T_{\bar{r}}^{(0)}\right)_{\bar{r}}\quad$
(121) $\displaystyle+\frac{\sigma^{(0)}\kappa^{(0)}}{2\bar{r}}\alpha
z_{b}^{(0)}\mathcal{G}^{(0)}I\;T^{(0)}_{\bar{r}}$

Thus, it becomes clear that the left hand side can be written as

$\displaystyle
w^{(0)}D_{s}\left[T_{c}^{(1)}-T^{(0)}_{\mathcal{M}^{(0)}}\mathcal{M}_{c}^{(1)}\right]-\frac{\bar{r}}{2}T_{\bar{r}}^{(0)}\dot{\mathbf{X}}^{(0)}_{s}\cdot\hat{\tau}$
(122)
$\displaystyle\qquad\qquad+\frac{\sigma^{(0)}\kappa^{(0)}}{2\bar{r}}\alpha
z_{b}^{(0)}\mathcal{G}^{(0)}\;I\;T_{\bar{r}}^{(0)}$

eventually yielding

$\displaystyle\int_{\mathcal{S}}^{[0,s]}\Bigg{[}-T_{t}^{(0)}+\frac{\bar{\eta}}{\bar{r}}\left(\bar{r}T_{\bar{r}}^{(0)}\right)_{\bar{r}}\hskip
85.35826pt$ (123)
$\displaystyle+\frac{\bar{r}}{2\sigma^{(0)}}T^{(0)}_{\bar{r}}\dot{\mathbf{X}}^{(0)}_{s}\cdot\hat{\tau}^{(0)}\Bigg{]}\frac{\sigma^{(0)}ds^{\prime}}{w^{(0)}}=0$

upon integration over a streamline $\mathcal{S}$, and assuming that the
filament satisfies the symmetry condition.
Thus, we have a complete set of equations for the motion of a buoyant vortex
filament, namely (67a) and (67b) together with the compatibility equations
(118), (123), (120).

Currently, the variables in (123), (120) and (118) are still given in terms of
$(t,\bar{r},s)$. We wish to write them in terms on the von Mises variables
$(t,\bar{M},s)$, in which $\mathcal{G}^{(0)}$, $\mathcal{H}^{(0)}$ and
$T^{(0)}$ are $s$ independent. We can replace $\mathcal{G}^{(0)}_{t}$ by
$\bar{\mathcal{G}}_{t}+\mathcal{G}_{\bar{M}}\bar{M}_{t}$ and the integral of
the first term in (118) by

$\bar{\mathcal{G}}_{t}\int_{\mathcal{S}}^{[0,s]}\frac{\sigma^{(0)}}{w^{(0)}}ds^{\prime}+\bar{\mathcal{G}}_{\bar{M}}\int_{\mathcal{S}}^{[0,s]}\mathcal{M}_{t}^{(0)}\frac{\sigma^{(0)}}{w^{(0)}}ds^{\prime}$
(124)

to arrive at a differential equation for $\bar{\mathcal{G}}$. Here we have
defined $\bar{f}$ as
$f(\bar{r},s,t)=\bar{f}({\cal{M}}^{(0)}(\bar{r},s,t),s,t)$ to distinguish
$f_{t}$ from $D_{t}(f)=\bar{f}_{t}$. We write $\bar{\mathcal{G}}_{\bar{M}}$ as
above, even if there is no ambiguity for this derivative. We should note that
in the case of horizontally placed filaments, the gravity terms drop out, and
(118) and (120) reduce to those of Klein and Ting (1992). Furthermore, in this
case one would be permitted to assume axial independence of the leading order
field components, which would result in (118) and (120) reducing to those of
Callegari and Ting (1978).

##  V Conclusions

In summary, the equation of motion, valid to first order, is

$\displaystyle\dot{\mathbf{X}}^{(0)}=\mathbf{Q}_{0}^{\perp}$ $\displaystyle+$
$\displaystyle\frac{\Gamma\kappa^{(0)}}{4\pi}\left[\ln\frac{1}{\epsilon}+C_{v}(s,t)+C_{w}(s,t)\right]\hat{\mathbf{b}}^{(0)}$
(125) $\displaystyle-$ $\displaystyle\frac{\Gamma}{4\pi}\alpha
C_{T}(s,t)\hat{\mathbf{z}}\times\hat{\mathbf{\tau}}^{(0)}$

where $C_{v}(s,t)$, $C_{w}(s,t)$ and $C_{T}(s,t)$ are given by (64), (65) and
(66), while $\alpha$ and $\hat{\mathbf{z}}$ are defined in (13). The equation
of motion couples to the compatibility conditions, (118), (120) and (123), to
form a complete set of equations for the motion of a buoyant vortex filament
assuming non-vanishing leading order axial velocity. Equation (125) is
equivalent to the equation obtained by Chang and Smith (2018) for a filament
with continuous density variation within the core, excluding higher order
terms. The perhaps most interesting feature of (125) is that it now contains a
contribution in the normal direction arising from the motion inside the core.
It is this term that results in the expansion of buoyant vortex rings as they
rise, as was demonstrated by Chang and Smith (2018). The binormal part of the
additional bouyancy term appearing in (125), is responsible for the tendency
of bouyant vortex rings to align themselves in the horizontal plane. Once
aligned in the horizontal plane gravity, by the compatibility conditions,
ceases to have an effect on the leading order total head, and only enters
directly in the core constant. Furthermore, the introduction of gravity at
first order necessarily imposes the condition of axial dependence on the
leading order field components. For a nearly straight, vertically aligned
vortex filament the influence of gravity will largely be indirectly through
the compatability conditions, in particular through its influence on the
leading order tangential velocity component. However, once it is slightly
tilted it will also necessarily have another contribution coming from the core
constant. This contribution would become more significant the greater the
tilt. Both of these contributions would in addition have to compete with the
logarithmic term in (125), which for truly thin filaments can be much larger.
Note, however, that for say, $\epsilon=0.1$, for which the asymptotic
predictions are relatively accurate according to Klein and Knio (1995), we
have $\ln(1/\epsilon)\sim 2$, so that the logarithmic term does not clearly
dominate the terms of order unity.

The next step would be to allow the background density to vary with height
$z$, following the anelastic approximation. One could also go one step further
and employ the pseudo-incompressible approximation for low Mach number flows.
We are certain such an analysis would yield new and interesting results,
bringing the theory closer to an accurate description of the dynamics of
tornado-like vorticies.

###### Acknowledgements.

This research was conducted while Marie Rodal was employed at FB Mathematik
and Informatik at Freie Univeristät Berlin, and was funded by the Deutsche
Forschungsgemeinschaft (DFG) through grant CRC 1114 "Scaling Cascades in
Complex Systems", Project Number 235221301, Project B07: Self-similar
structures in turbluent flows and the construction of LES closures. Marie
Rodal additionally acknowledges with gratitude the support of the Research
Fund of the University of Antwerp.

Finally, we thank the developers of SageMath(The Sage Developers, 2023), an
open source computer algebra system (or symbolic calculator) with which we
performed symbolic calculation to cross check and chase typos and errors in
such a complex calculation.

## Code Availability

The SageMath code containing the calculations of the preceding sections can be
found in the dedicated GitHub repository at
https://github.com/danielmargerit/SageMAE4PDEs.

## Declaration of interests

The authors report no conflict of interest.

##  VI Appendix

##  Appendix A Computation of core constant integral

We wish to simplify the integral

$\displaystyle
C_{0}(s,t)=\lim_{\bar{r}\longrightarrow\infty}\Bigg{[}\frac{v^{(0)}\kappa^{(0)}}{\bar{r}}\int_{0}^{\bar{r}}\frac{1}{2\left(v^{(0)}\right)^{2}}\int_{0}^{z}\left[\xi
v^{(0)}H(\xi,t)d\xi\right]dz$
$\displaystyle-\frac{\Gamma}{4\pi}\ln\bar{r}\Bigg{]}\qquad$ (126)

where

$H(\xi,t)=2\xi\zeta^{(0)}+v^{(0)}+\frac{2\xi
w^{(0)}w^{(0)}_{\bar{r}}}{v^{(0)}}$ (127)

and the asymptotic behaviour of $v^{(0)}$, $\zeta^{(0)}$ and $w^{(0)}$ are
given by (53a), (53b) and (46). We note that this calculation was first
carried out by Callegari and Ting (1978), but as these authors did not
explicitly include all the details of the calculation, we are including them
here to avoid any potential confusion.

To simplify, we write

$C_{0}(s,t)=\kappa^{(0)}\lim_{\bar{r}\longrightarrow\infty}\left[\frac{v^{(0)}}{\bar{r}}h(\bar{r},t)-\frac{\Gamma}{4\pi}\ln\bar{r}\right]$
(128)

where we have defined

$h(\bar{r},t)=\int_{0}^{\bar{r}}\frac{1}{z\left(v^{(0)}\right)^{2}}\int_{0}^{z}\left[\xi
v^{(0)}H(\xi,t)d\xi\right]dz$ (129)

Consequently, we get

$\displaystyle C_{0}(s,t)$ $\displaystyle=$
$\displaystyle\kappa^{(0)}\lim_{\bar{r}\longrightarrow\infty}\left[\frac{v^{(0)}\bar{r}}{\bar{r}^{2}}h(\bar{r},t)-\frac{\Gamma}{4\pi}\ln\bar{r}\right]$
(130) $\displaystyle=$
$\displaystyle\kappa^{(0)}\lim_{\bar{r}\longrightarrow\infty}\left[\frac{\Gamma}{2\pi\bar{r}^{2}}h(\bar{r},t)-\frac{\Gamma}{4\pi}\ln\bar{r}\right]$
(131) $\displaystyle=$
$\displaystyle\frac{\Gamma}{2\pi}\kappa^{(0)}\lim_{\bar{r}\longrightarrow\infty}\left[\frac{h(\bar{r},t)-\frac{\bar{r}^{2}}{2}\ln\bar{r}}{\bar{r}^{2}}\right]$
(132)

Application of the l’Hopital’s rule, yields

$\displaystyle
C_{0}(s,t)=\frac{\Gamma}{2\pi}\kappa^{(0)}\lim_{\bar{r}\longrightarrow\infty}\left[\frac{\left(h-\frac{\bar{r}^{2}}{2}\ln\bar{r}\right)^{\prime}}{\left(\bar{r}^{2}\right)^{\prime}}\right]\qquad\qquad\qquad\qquad$
$\displaystyle=\frac{\Gamma}{2\pi}\kappa^{(0)}\lim_{\bar{r}\longrightarrow\infty}\left[\frac{\frac{1}{\bar{r}\left(v^{(0)}\right)^{2}}\int_{0}^{\bar{r}}\left[\xi
v^{(0)}H(\xi,t)d\xi\right]-\frac{\bar{r}}{2}-\bar{r}\ln\bar{r}}{2\bar{r}}\right]$
$\displaystyle=\frac{\Gamma}{4\pi}\kappa^{(0)}\lim_{\bar{r}\longrightarrow\infty}\left[\frac{4\pi^{2}}{\Gamma^{2}}\int_{0}^{\bar{r}}\xi
v^{(0)}H(\xi,t)d\xi-\frac{1}{2}-\ln{\bar{r}}\right]\qquad\;\;$
$\displaystyle=\frac{\Gamma}{4\pi}\kappa^{(0)}\lim_{\bar{r}\longrightarrow\infty}\left[\int_{0}^{\bar{r}}\xi
v^{(0)}H(\xi,t)d\xi-\frac{1}{2}-\ln\bar{r}\right]\qquad\qquad\;\;$

Next, we note that

$\displaystyle\lim_{\bar{r}\longrightarrow\infty}\int_{0}^{\bar{r}}\xi
v^{(0)}\left(\frac{2\xi
w^{(0)}w^{(0)}_{\bar{r}}}{v^{(0)}}\right)d\xi\qquad\qquad\qquad\qquad$ (133)
$\displaystyle\
=\lim_{\bar{r}\longrightarrow\infty}2\int_{0}^{\bar{r}}\xi^{2}w^{(0)}w^{(0)}_{\bar{r}}d\xi\qquad\qquad\qquad\quad$
(134)
$\displaystyle=\lim_{\bar{r}\longrightarrow\infty}\int_{0}^{\bar{r}}\xi^{2}\frac{\partial}{\partial\xi}\left(w^{(0)}\right)^{2}d\xi\qquad\qquad\qquad$
(135)
$\displaystyle=\left[\xi^{2}\left(w^{(0)}\right)^{2}\right]_{0}^{\infty}-\int_{0}^{\infty}\left(w^{(0)}\right)^{2}2\xi
d\xi\quad\quad$ (136)
$\displaystyle=-2\int_{0}^{\infty}\left(w^{(0)}\right)^{2}\xi
d\xi\;\qquad\qquad\qquad\qquad$ (137)

Similarly, we have

$\displaystyle\int_{0}^{\bar{r}}2\xi^{2}v^{(0)}\zeta^{(0)}d\xi$
$\displaystyle=$ $\displaystyle\int_{0}^{\bar{r}}2\xi v^{(0)}\left(\xi
v^{(0)}\right)_{\xi}d\xi$ (138) $\displaystyle=$
$\displaystyle\int_{0}^{\bar{r}}\frac{\partial}{\partial\xi}\left(\xi
v^{(0)}\right)^{2}d\xi$ (139) $\displaystyle=$ $\displaystyle\left[\left(\xi
v^{(0)}\right)^{2}\right]^{\bar{r}}_{0}$ (140) $\displaystyle=$
$\displaystyle\left(\bar{r}v^{(0)}\right)^{2}$ (141)

and thus,

$\displaystyle\lim_{\bar{r}\longrightarrow\infty}\frac{4\pi^{2}}{\Gamma^{2}}\left(\int_{0}^{\bar{r}}2\xi^{2}v^{(0)}\zeta^{(0)}d\xi\right)$
$\displaystyle=$
$\displaystyle\lim_{\bar{r}\longrightarrow\infty}\frac{4\pi^{2}}{\Gamma^{2}}\left(\bar{r}v^{(0)}\right)^{2}$
$\displaystyle=$
$\displaystyle\frac{4\pi^{2}}{\Gamma^{2}}\left(\frac{\Gamma}{2\pi}\right)^{2}$
$\displaystyle=$ $\displaystyle 1$

Consequently, equation $(\ref{appendix_core_constant})$ can be written as

$\displaystyle
C_{0}(s,t)=\frac{\Gamma}{4\pi}\kappa^{(0)}\Bigg{[}\lim_{\bar{r}\longrightarrow\infty}\left(\frac{4\pi^{2}}{\Gamma^{2}}\int_{0}^{\bar{r}}\xi\left(v^{(0)}\right)^{2}d\xi-\ln\bar{r}\right)\qquad$
(142)
$\displaystyle+\frac{1}{2}-\frac{8\pi^{2}}{\Gamma^{2}}\int_{0}^{\infty}\left(w^{(0)}\right)^{2}\xi
d\xi\Bigg{]}\qquad$

##  Appendix B The kinetic energy equation

The Navier Stokes equation in Cartesian coordinates given the Boussinesq
approximation is

$\left(\frac{\partial\mathbf{v}}{\partial
t}\right)_{xyz}+\mathbf{v}\cdot\nabla_{xyz}\mathbf{v}=-\nabla_{xyz}P+\bar{\nu}\epsilon^{2}\Delta_{xyz}\mathbf{v}-\epsilon^{-2}\alpha\tilde{T}\hat{\mathbf{z}}$
(143)

where we use the subscript $xyz$ to indicate that these are derivatives with
respect to the Cartesian coordinate system.
The kinetic energy equation is obtained by contracting both sides with
$\mathbf{v}$, yielding

$\displaystyle\left(\frac{\partial(\mathbf{v}^{2}/2)}{\partial
t}\right)_{xyz}=-\mathbf{v}\cdot\nabla_{xyz}\left[\mathbf{v}^{2}/2+P\right]+\bar{\nu}\epsilon^{2}\mathbf{v}\cdot\Delta_{xyz}\mathbf{v}\qquad$
(144)
$\displaystyle-\epsilon^{-2}\alpha\tilde{T}\mathbf{v}\cdot\hat{\mathbf{z}}$

where we have used

$\displaystyle\nabla_{xyz}(\mathbf{v}^{2}/2)$ $\displaystyle=$
$\displaystyle\mathbf{v}\cdot\nabla_{xyz}\mathbf{v}+\mathbf{v}\times\mathbf{w}$
(145)

and

$\mathbf{v}\cdot\left(\mathbf{v}\times\mathbf{w}\right)=0$ (146)

We can write (144) in terms of the total head, $\mathcal{H}$, in Cartesian
coordinates, as follows

$\displaystyle\left(\frac{\partial\mathcal{H}}{\partial
t}\right)_{xyz}+\mathbf{v}\cdot\nabla_{xyz}\mathcal{H}=\left(\frac{\partial
P}{\partial
t}\right)_{xyz}+\epsilon^{2}\bar{\nu}\mathbf{v}\cdot\Delta_{xyz}\mathbf{v}$
(147) $\displaystyle+\epsilon^{-2}\alpha
T\left(\frac{\partial\mathcal{Z}}{\partial
t}\right)_{xyz}+\alpha\bar{\eta}\mathcal{Z}\Delta_{xyz}T$

where

$\mathcal{H}=P+\mathbf{v}^{2}/2+\epsilon^{-2}\alpha\tilde{T}\mathcal{Z}$ (148)

$\mathcal{Z}=z-z_{0}$ is the height and

$\nabla_{xyz}\mathcal{Z}=\hat{\mathbf{z}}$ (149)

In deriving (147) we have used the temperature equation, which in Cartesian
coordinates is

$\left(\frac{\partial T}{\partial
t}\right)_{xyz}+\mathbf{v}\cdot\nabla_{xyz}T=\epsilon^{2}\bar{\eta}\Delta_{xyz}T$
(150)

to remove the time derivatives of the temperature $T$. In curvilinear
coordinates (147) becomes

$\displaystyle\left(\frac{\partial\mathcal{H}}{\partial t}\right)_{r\varphi
s}+\mathbf{v}_{R}\cdot\nabla_{r\varphi s}\mathcal{H}=\left(\frac{\partial
P}{\partial t}\right)_{r\varphi s}-\mathbf{v}_{E}\cdot\nabla_{r\varphi s}P$
$\displaystyle+\epsilon^{2}\bar{\nu}\left[\dot{\mathbf{X}}(s,t)+\mathbf{V}\right]\cdot\Delta_{r\varphi
s}\left[\dot{\mathbf{X}}(s,t)+\mathbf{V}\right]+\alpha\bar{\eta}\mathcal{Z}\Delta_{r\varphi
s}T$ (151) $\displaystyle+\epsilon^{-2}\alpha
T\left(\frac{\partial\mathcal{Z}}{\partial t}\right)_{r\varphi
s}-\epsilon^{-2}\alpha T\mathbf{v}_{E}\cdot\nabla_{r\varphi s}\mathcal{Z}$

where

$\displaystyle\mathbf{v}_{R}$ $\displaystyle=$
$\displaystyle\mathbf{V}-r\left(\frac{\partial\hat{\mathbf{r}}}{\partial
t}\right)_{r\varphi s}$ (152) $\displaystyle\mathbf{v}_{E}$ $\displaystyle=$
$\displaystyle\dot{\mathbf{X}}+r\left(\frac{\partial\hat{\mathbf{r}}}{\partial
t}\right)_{r\varphi s}$ (153)

and

$\displaystyle\mathcal{H}=P+\left(\dot{\mathbf{X}}(s,t)+\mathbf{V}\right)^{2}/2+\epsilon^{-2}\alpha\tilde{T}\mathcal{Z}$
(154)
$\displaystyle\mathcal{Z}(\bar{r},\varphi,s,t)=\left[\mathbf{X}(s,t)+r\hat{\mathbf{r}}-\mathbf{X}(0,t)\right]\cdot\hat{\mathbf{z}}$
(155)

To transform equation (144) we have used

$\displaystyle\left(\frac{\partial\mathcal{H}}{\partial
t}\right)_{xyz}+\mathbf{v}\cdot\nabla_{xyz}\mathcal{H}=\left(\frac{\partial\mathcal{H}}{\partial
t}\right)_{r\varphi s}+\mathbf{v}_{R}\cdot\nabla_{r\varphi
s}\mathcal{H}\qquad\qquad$ (156) $\displaystyle\left(\frac{\partial
P}{\partial t}\right)_{xyz}=\left(\frac{\partial P}{\partial
t}\right)_{r\varphi s}-\mathbf{v}_{E}\cdot\nabla_{r\varphi s}P\qquad$ (157)
$\displaystyle\left(\frac{\partial\mathcal{Z}}{\partial
t}\right)_{xyz}=\left(\frac{\partial\mathcal{Z}}{\partial t}\right)_{r\varphi
s}-\mathbf{v}_{E}\cdot\nabla_{r\varphi s}\mathcal{Z}\qquad$ (158)

Here $\nabla_{r\varphi s}\mathcal{Z}=\mathbf{\hat{z}}$. More details on the
coordinate transformation can be found in Callegari and Ting (1978), Appendix
A and B.

The leading and second order equations of (151) are, respectively

$u^{(1)}\mathcal{H}^{(0)}_{\bar{r}}+\frac{v^{(0)}}{\bar{r}}\mathcal{H}_{\varphi}^{(1)}+\frac{\omega^{(0)}}{\sigma^{(0)}}\mathcal{H}_{s}^{(0)}=-\dot{\mathbf{X}}^{(0)}\cdot\hat{\mathbf{r}}^{(0)}P^{(0)}_{\bar{r}}$
(159)

and

$\displaystyle\mathcal{H}_{t}^{(0)}+u^{(2)}\mathcal{H}^{(0)}_{\bar{r}}+u^{(1)}\mathcal{H}^{(1)}_{\bar{r}}+\frac{v^{(1)}}{\bar{r}}\mathcal{H}^{(1)}_{\varphi}$
$\displaystyle+\frac{w^{(1)}}{\sigma^{(0)}}\mathcal{H}^{(0)}_{s}+\frac{v^{(0)}}{\bar{r}}\mathcal{H}^{(2)}_{\varphi}+\frac{w^{(0)}}{\sigma^{(0)}}\mathcal{H}^{(1)}_{s}-\frac{w^{(0)}\sigma^{(1)}}{\left(\sigma^{(0)}\right)^{2}}\mathcal{H}^{(0)}_{s}$
$\displaystyle=P_{t}^{(0)}-\left(\dot{\mathbf{X}}\cdot\hat{\mathbf{r}}\right)^{(1)}P_{\bar{r}}^{(0)}-\dot{\mathbf{X}}^{(0)}\cdot\hat{\mathbf{r}}^{(0)}P^{(1)}_{\bar{r}}$
$\displaystyle-\frac{1}{\bar{r}}\dot{\mathbf{X}}^{(0)}\cdot\hat{\theta}^{(0)}P^{(1)}_{\varphi}+\alpha\tilde{T}^{(0)}\mathcal{Z}^{(0)}_{t}+\bar{\nu}V^{(0)}\Delta
V^{(0)}$ (160)
$\displaystyle-\alpha\tilde{T}^{(0)}\mathbf{v}_{E}^{(0)}\cdot\mathbf{\hat{z}}+\alpha\bar{\eta}\mathcal{Z}^{(0)}\Delta\tilde{T}^{(0)}$

where we have dropped the subscript ($\bar{r}\varphi s$) for notational
convenience. Averaging with respect to $\varphi$ yields the leading and first
order symmetric equations:

$\frac{w^{(0)}}{\sigma^{(0)}}\mathcal{H}^{(0)}_{s}+u_{c}^{(1)}\mathcal{H}_{\bar{r}}^{(0)}=0$
(161)

and

$\displaystyle\mathcal{H}^{(0)}_{t}+u_{c}^{(2)}\mathcal{H}^{(0)}_{\bar{r}}+u_{c}^{(1)}\left(\mathcal{H}^{(1)}_{c}\right)_{\bar{r}}+\frac{w^{(1)}_{c}}{\sigma^{(0)}}\mathcal{H}_{s}^{(0)}+$
$\displaystyle\frac{w^{(0)}}{\sigma^{(0)}}\left(\mathcal{H}_{c}^{(1)}\right)_{s}-\frac{w^{(0)}\sigma^{(1)}}{\left(\sigma^{(0)}\right)^{2}}\mathcal{H}^{(0)}_{s}+\frac{1}{2}\left(\mathcal{H}_{11}^{(0)}\right)u^{(1)}_{11}$
$\displaystyle+\frac{1}{2}\left(\mathcal{H}_{12}^{(1)}\right)_{\bar{r}}u^{(1)}_{12}-\frac{1}{2\bar{r}}\mathcal{H}_{11}^{(1)}v_{12}^{(1)}+\frac{1}{2\bar{r}}\mathcal{H}_{12}^{(1)}v_{11}^{(1)}$
(162)
$\displaystyle=P_{t}^{(0)}-\frac{1}{2}\dot{\mathbf{X}}^{(0)}\cdot\hat{\mathbf{n}}^{(0)}\left(P^{(1)}_{11}\right)_{\bar{r}}-\frac{1}{2}\dot{\mathbf{X}}^{(0)}\cdot\hat{\mathbf{b}}^{(0)}\left(P_{12}^{(1)}\right)_{\bar{r}}$
$\displaystyle-\frac{1}{2\bar{r}}\dot{\mathbf{X}}^{(0)}\cdot\hat{\mathbf{b}}^{(0)}P_{12}^{(1)}-\frac{1}{2\bar{r}}\dot{\mathbf{X}}^{(0)}\cdot\hat{\mathbf{n}}^{(0)}P_{11}^{(1)}-\alpha\tilde{T}^{(0)}\dot{\mathbf{X}}^{(0)}\cdot\hat{\mathbf{z}}$
$\displaystyle+\alpha\tilde{T}^{(0)}\left(\mathcal{Z}_{c}^{(0)}\right)_{t}+\bar{\nu}V^{(0)}\Delta
V^{(0)}+\alpha\bar{\eta}\mathcal{Z}_{c}^{(0)}\Delta\tilde{T}^{(0)}$

Here we have written the asymmetric part of $\mathcal{H}^{(1)}$ as

$\displaystyle\mathcal{H}_{a}^{(1)}=\mathcal{H}^{(1)}_{11}\cos\varphi+\mathcal{H}^{(1)}_{12}\sin\varphi$
(163)

where

$\displaystyle\mathcal{H}_{11}^{(1)}=P_{11}^{(1)}+v^{(0)}v_{11}^{(1)}+w^{(0)}w_{11}^{(1)}\qquad\qquad$
(164)
$\displaystyle\qquad\qquad+\alpha\mathcal{Z}_{c}^{(0)}T_{11}^{(1)}+\alpha\bar{r}\tilde{T}^{(0)}z_{\text{n}}^{(0)}+v^{(0)}\dot{\mathbf{X}}^{(0)}\cdot\hat{\mathbf{b}}^{(0)}$
$\displaystyle\mathcal{H}_{12}^{(1)}=P_{12}^{(1)}+v^{(0)}v_{12}^{(1)}+w^{(0)}w_{12}^{(1)}\qquad\qquad$
(165)
$\displaystyle\qquad\qquad+\alpha\mathcal{Z}_{c}^{(0)}T_{12}^{(1)}+\alpha\bar{r}\tilde{T}^{(0)}z_{\text{b}}^{(0)}+v^{(0)}\dot{\mathbf{X}}^{(0)}\cdot\hat{\mathbf{b}}^{(0)}$

We additionally note that, using (44b) we can write $P_{11}^{(1)}$ and
$P_{12}^{(1)}$ as

$\displaystyle
P_{11}=u_{12}^{(1)}\bar{r}v_{\bar{r}}^{(0)}-v^{(0)}v_{11}^{(1)}+v^{(0)}u^{(1)}_{12}\qquad\qquad\qquad$
(166)
$\displaystyle\qquad\qquad\qquad-\left(w^{(0)}\right)^{2}\kappa^{(0)}-\alpha\bar{r}\tilde{T}^{(0)}z_{\text{n}}^{(0)}$
$\displaystyle
P_{12}=-u_{11}^{(1)}\bar{r}v_{\bar{r}}^{(0)}-v^{(0)}u_{11}^{(1)}-v^{(0)}v_{12}^{(1)}-\alpha\bar{r}\tilde{T}^{(0)}z_{\text{b}}^{(0)}$
(167)

Using these relations, (B23) can be rewritten to eventually yield (102).

##  Appendix C The case with no leading order axial velocity

When deriving the evolution equations in section IV, the assumption was made
that $w^{(0)}\neq 0$ at any point in the vortex core. This allowed us to
introduce the s-derivative along stream lines, and thus conclude that along
stream lines the leading order circulation, total head and temperature are all
s-independent on these curves. However, the case when $w^{(0)}=0$ at any point
in the vortex core requires special attention, which we will briefly cover in
this section. We will not treat here the special case of a stagnation point
where $w^{(0)}=0$ at a single point.

Whenever $w^{(0)}=0$ we get, from (44a), (80a) and (38)

$w_{a}^{(1)}=0\qquad\text{ and }\qquad u_{c}^{(1)}=0$ (168)

Using this in (81a) and (85) yields

$\displaystyle P_{s}^{(0)}$ $\displaystyle=$
$\displaystyle-\alpha\sigma^{(0)}\tilde{T}^{(0)}z_{\text{t}}^{(0)}$ (169)
$\displaystyle P_{s}^{(1)}$ $\displaystyle=$
$\displaystyle-\alpha\sigma^{(0)}\left(\tilde{T}^{(1)}_{c}z_{\text{t}}^{(0)}+\tilde{T}^{(0)}z_{t}^{(1)}+\frac{\sigma^{(1)}}{\sigma^{(0)}}\tilde{T}^{(0)}z_{\text{t}}^{(0)}\right)\qquad$
(170)

Thus, if $P_{s}^{(0)}=0$, then we must necessarily have that
$\tilde{T}^{(0)}=0$ and we revert back to the case described by Callegari and
Ting (1978) with no gravitational force and no leading order axial variation.

Assuming $w^{(0)}=0$ and $P_{s}^{(0)}\neq 0$, the symmetric second order
circumferential momentum and temperature equations are, respectively

$\displaystyle
v_{t}^{(0)}+u_{c}^{(2)}\frac{1}{\bar{r}}(\bar{r}v^{(0)})_{\bar{r}}+\frac{w_{c}^{(1)}v_{s}^{(0)}}{\sigma^{(0)}}\qquad\qquad$
$\displaystyle=\frac{\bar{\nu}}{\bar{r}}\left(\bar{r}v^{(0)}_{\bar{r}}\right)_{\bar{r}}-\frac{\bar{\nu}}{\bar{r}}v^{(0)}+\frac{\alpha}{2}\kappa^{(0)}z_{b}^{(0)}v^{(0)}I\zeta^{(0)}$
(171)

and

$\displaystyle
T_{t}^{(0)}+u_{c}^{(2)}T^{(0)}_{\bar{r}}+\frac{w_{c}^{(1)}}{\sigma^{(0)}}T_{s}^{(0)}\qquad\qquad$
$\displaystyle=\frac{\bar{\eta}}{\bar{r}}\left(\bar{r}T_{\bar{r}}^{(0)}\right)_{\bar{r}}+\frac{\alpha}{2}\kappa^{(0)}z_{b}^{(0)}v^{(0)}IT_{\bar{r}}^{(0)}$
(172)

From the second order continuity equation, we additionally have

$\displaystyle
u_{c}^{(2)}=-\frac{\bar{r}}{2\sigma^{(0)}}\left(\dot{\mathbf{X}}_{s}^{(0)}\cdot\hat{\tau}^{(0)}\right)+\frac{\kappa^{(0)}}{2}\alpha
z_{b}^{(0)}v^{(0)}I\qquad\qquad$
$\displaystyle-\frac{1}{\bar{r}\sigma^{(0)}}\int_{0}^{\bar{r}}r^{\prime}\left(w_{c}^{(1)}\right)_{s}dr^{\prime}$
(173)

Contrary to the case without axial variation, here $w_{c}^{(1)}$ appears in
those leading order equation for the core structure dynamical evolution, which
means that an equation for $w_{c}^{(1)}$ is needed to close the equations. The
third order symmetric tangential momentum equation yields the temporal
evolution equation for $w_{c}^{(1)}$:

$\displaystyle(w_{c}^{(1)})_{t}+g_{4}^{(3)}w_{c}^{(1)}+u_{c}^{(2)}\left(w_{c}^{(1)}\right)_{\bar{r}}+\frac{w_{c}^{(1)}}{\sigma^{(0)}}\left(w_{c}^{(1)}\right)_{s}$
$\displaystyle=-\frac{\left(P_{c}^{(2)}\right)_{s}}{\sigma^{(0)}}+\frac{\sigma^{(1)}}{\left(\sigma^{(0)}\right)^{2}}\left(P_{c}^{(1)}\right)_{s}+\frac{\bar{\nu}}{\bar{r}}\left(\bar{r}\left(w_{c}^{(1)}\right)_{\bar{r}}\right)_{\bar{r}}$
(174)
$\displaystyle-\alpha\left(\tilde{T}_{c}^{(2)}z_{\text{t}}^{(0)}+\tilde{T}_{c}^{(1)}z_{\text{t}}^{(1)}+\tilde{T}^{(0)}z_{\text{t}}^{(2)}\right)+s_{4}^{(0)}$

where we have defined

$\displaystyle
g_{4}^{(3)}=\frac{\dot{\mathbf{X}}^{(0)}_{s}\cdot\hat{\tau}^{(0)}}{\sigma^{(0)}}-\frac{\kappa^{(0)}}{2}\left(u_{11}^{(1)}-v_{12}^{(1)}\right)$
(175)

and

$\displaystyle
s_{4}^{(3)}=-\ddot{\mathbf{X}}^{(0)}\cdot\hat{\tau}^{(0)}-\kappa^{(0)}v^{(0)}\left<w_{a}^{(2)}\sin\varphi\right>-\frac{\kappa^{(0)}\bar{r}}{2\sigma^{(0)}}\left(P_{11}^{(1)}\right)_{s}$
$\displaystyle-\frac{1}{2}\frac{\partial\hat{\mathbf{n}}^{(0)}}{\partial
t}\cdot\hat{\tau}^{(0)}\left(u_{11}^{(1)}-v_{12}^{(1)}\right)-\frac{1}{\bar{r}}\left<v_{a}^{(1)}\left(w_{a}^{(2)}\right)_{\varphi}\right>$
(176) $\displaystyle-\frac{1}{2}\frac{\partial\hat{\mathbf{b}}^{(0)}}{\partial
t}\cdot\hat{\tau}^{(0)}\left(u_{12}^{(1)}+v_{11}^{(1)}-v^{(0)}\kappa^{(0)}\bar{r}\right)-\left<u_{a}^{(1)}\left(w_{a}^{(2)}\right)_{\bar{r}}\right>$
$\displaystyle+\frac{\sigma^{(2)}\sigma^{(0)}-\left(\sigma^{(1)}\right)^{2}-\frac{1}{2}\left(\kappa^{(0)}\sigma^{(0)}\bar{r}\right)^{2}}{\left(\sigma^{(0)}\right)^{3}}P_{s}^{(0)}-\frac{\bar{r}\kappa^{(0)}\mathcal{T}_{\text{o}}^{(0)}}{2}P_{12}^{(1)}$

Here we have introduced the notation $\left<...\right>$ to indicate averaging
w.r.t. $\varphi$. The $P_{c}^{(2)}$-term in (C7) can be removed by averaging
over $s$, while $w_{a}^{(2)}$ can be found from the asymmetric part of the 2nd
order tangential momentum equation (not shown). However, $P_{c}^{(1)}$ is
related to $v_{c}^{(1)}$ through (81c), and thus a fourth evolution equation,
for $v_{c}^{(1)}$, is required. This equation is given by the symmetric 3rd
order circumferential momentum equation (not shown). The complete derivation
lies outside the scope of this work, as it will require the higher order
analysis (i.e. derivation of expressions for $u_{a}^{(2)}$, $v_{a}^{(2)}$ and
$\dot{\mathbf{X}}^{(1)}$). However, in principle the derivation would proceed
analogously to the case presented in section IV, with introduction of
appropriate derivatives along stream lines defined by the second order
continuity equation. Concretely, if $w_{c}^{(1)}\neq 0$ we can introduce the
s-derivative along stream lines defined by the leading order axial mass flux,
$\mathcal{M}^{(1)}_{c}$. This derivative is given as

$\displaystyle
D^{(1)}_{s}=\partial_{s}-\left(\mathcal{M}^{(1)}_{c}\right)_{s}\>/\>\left(\mathcal{M}^{(1)}_{c}\right)_{\bar{r}}\partial_{\bar{r}}\qquad\qquad\qquad$
(177)
$\displaystyle=\partial_{s}+\left(\frac{\sigma^{(0)}\bar{r}u_{c}^{(2)}}{w_{c}^{(1)}}+\frac{\bar{r}}{2}\frac{\left(\dot{\mathbf{X}}^{(0)}_{s}\cdot\hat{\tau}^{(0)}\right)}{w_{c}^{(1)}}-\frac{\sigma^{(0)}\kappa^{(0)}}{2w_{c}^{(1)}}\alpha
z_{b}^{(0)}v^{(0)}I\right)\partial_{\bar{r}}$

where we have used the superscript $(1)$ to distinguish this derivative from
the $D_{s}$ derivative introduced in section IV.

This brief discussion highlights the importance of distinguishing between the
three distinct cases of (1) non-vanishing leading order axial velocity
($w^{(0)}\neq 0$), (2) vanishing leading and first order axial velocity
($w^{(0)}=0$ and $w^{(1)}=0$) and (3) the intermediate case of vanishing
leading order axial velocity ($w^{(0)}=0$ and $w^{(1)}\neq 0$).

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