# On the Measurement of Vorticity in Astrophysical Fluids

Steven R. Spangler Department of Physics and Astronomy, University of Iowa

###### Abstract

Vorticity is central to the nature of, and dynamical processes in turbulence,
including turbulence in astrophysical fluids. The results of Raymond et al
(2020a, b) on vorticity in the post-shock fluid of the Cygnus Loop supernova
remnant are therefore of great interest. We consider the degree to which
spectroscopic measurements of an optically-thin line, the most common type of
astronomical velocimetry, can yield unambiguous measurements of the vorticity
in a fluid. We consider an ideal case of observations in the plane of a flow
which may or may not contain vorticity. In one case, the flow possesses
vorticity in a direction perpendicular to the plane of observations. In the
other case, the flow is irrotational (zero vorticity) by construction. The
observationally-deduced vorticity (referred to as the pseudovorticity) is
inferred from spatial differences in the line-of-sight component of velocity,
and assumptions of symmetry. My principal result is that in the case of the
vortical flow, the pseudovorticity is a reasonable match for the true
vorticity. However, and importantly, the pseudovorticity in the case of the
irrotational flow field is also nonzero, and comparable in magnitude to that
for a vortical flow. The conclusion of this paper is that while astronomical
spectroscopic observations may yield a good estimate of the vorticity in a
remote fluid, the robustness of such an inference cannot be insured.

interstellar medium — turbulence — interstellar plasma

##  1 1\. Introduction

“Zeus no longer exists, a Vortex now rules in his place ” …Aristophanes, “The
Clouds”

Turbulence is believed to play an important, and perhaps dominant role in many
astrophysical processes. A full litany of such processes would take a great
deal of space, and has been given in many other publications. However, to cite
just one example, turbulence is believed to play an important role in
modulating star formation (Guszejnov et al, 2020). It may be important to
distinguish between turbulence in plasmas, for which the Lorentz force is a
dominant term, and turbulence in neutral fluids (hydrodynamic turbulence), in
which that term is absent. However, the most common theoretical description of
plasmas used in astrophysics, single fluid magnetohydrodynamics (MHD),
contains all the terms present in the Navier-Stokes equations. One therefore
expects close similarities between hydrodynamic and magnetohydrodynamic
turbulence.

Vorticity is crucial to the nature and dynamics of turbulence, a fact
apparently recognized by the Pre-Socratic philosophers of Ancient Greece, and
conveyed in the above excerpt from The Clouds of Aristophanes, written around
423 BCE 111This phrase was so important to the theme of “The Clouds”, that
Aristophanes included it three times in the comedy.. In terms of a modern
mathematical description of fluid mechanics, the vorticity $\vec{\omega}$ is
related to the fluid velocity field $\vec{v}(\vec{r},t)$ as

$\vec{\omega}=\nabla\times\vec{v}$ (1)

where $\vec{r}$ is the vector position and $t$ is the time.

Descriptions of hydrodynamic turbulence generally envision the turbulent field
as a superposition of vorticies of differing sizes and intensities (e.g
Bradshaw, 1978; Vincent and Meneguzzi, 1991). Furthermore, the amplification
of vorticity in the strain field

$S_{ij}\equiv\frac{1}{2}\left(\frac{\partial v_{i}}{\partial
x_{j}}+\frac{\partial v_{j}}{\partial x_{i}}\right)$ (2)

of the surrounding flow is considered the dominant mechanism for the famous
turbulent cascade from large scales to small scales. As clearly explained in
Bradshaw (1978) a vortex on one spatial scale is stretched and amplified by
the flow field of vorticies on larger scales.

It should be noted that this view of “vortex stretching” as the sole mechanism
causing the turbulent cascade may need revision. Recent work (Johnson, 2021;
Carbone & Bragg, 2020) has indicated that “strain amplification” can compete
with, and perhaps dominate, vortex stretching as a cascade process. Strain
amplification can simplistically be described as enhancement of gradients due
to convergent flows, and is familiar from one dimensional nonlinear wave
equations such as Burger’s Equation and the Derivative Nonlinear Schroedinger
Equation (e.g. Spangler, 1985). However, it may be the case that vorticity
dynamics are responsible for highly intermittent events in turbulence ( see
Johnson & Meneveau, 2016).

A variety of astronomical measurements have shown that turbulence exists in
media such as the Solar Corona, all phases of the Galactic Interstellar
Medium, and perhaps extragalactic radio sources. However, the diagnostics
available to astronomers, such as radio wave scattering from density
fluctuations (Rickett, 1990) or turbulently-broadened spectral lines (Redfield
and Linsky, 2004) are not suited for insight into the dynamics of the
turbulence. The case of the Solar Wind is in much better shape, in that
impressive, in-situ measurements of flow velocity, magnetic field, and density
fluctuations can be made over a wide range of spatial scales (Tu and Marsch,
1995; Bruno and Carbone, 2005). Even here, however, inferring quantities such
as the vorticity and determining the turbulent dynamics are very difficult
(see, however Andres et al, 2021).

Given the above remarks, it would seem virtually impossible to measure the
vorticity, rate-of-strain tensor, or other such quantity in remotely-sensed
astrophysical media. For this reason, the recent results of Raymond et al
(2020a, b) were a cause of great enthusiasm for astronomical “cumaphilonts”
222A Hellenistically-motivated term meaning “lovers of turbulence” that I
originally introduced in Spangler (2001), without any noticeable success..
Utilizing spectroscopic and proper motion measurements of post-shock flow in
the western portion of the Cygnus Loop, Raymond et al (2020a, b) report
detection and measurement of vorticity in that flow. Raymond et al (2020a, b)
were clear and candid in the assumptions that were used in their vorticity
detection. In particular, assumptions were made about the symmetry of the flow
that could not be subject to observational verification or falsification.

The results of Raymond et al (2020a, b) motivated the present investigation. I
investigate the spectroscopic inference of vorticity for a set of idealized,
model flows. I restrict attention to spectroscopic measurements of flow
Doppler shifts, because that is the most common method of astronomical
anemometry. The results presented in Raymond et al (2020a) used only
spectroscopically-determined Doppler shifts, and thus fall in this category.
Proper motion measurements as utilized by Raymond et al (2020b) are an
exception to this statement, as are some types of radio scintillation
measurements that measure the velocity at which a scintillation pattern moves
across a radio telescope array ( e.g. Coles et al, 1991). The goals of the
present investigation are to determine (1) if spatially-varying Doppler shifts
measured spectroscopically in the case of a true vortical flow allow an
accurate estimate of the vorticity to be retrieved, and (2) if flows with
$\vec{\omega}=0$ by construction would yield a spectroscopic signature which
would be misinterpreted as vorticity.

The organization of this paper is as follows. In Section 2, I make a cursory
inspection of laboratory measurements of vorticity in hydrodynamic turbulence.
The point of including this section, other than its intrinsic interest, is to
acknowledge that even in controlled laboratory settings with internal and
external probes, measurement of vorticity is difficult. Section 3 considers
the flow associated with a vortex being amplified by a strain along the vortex
axis, referred to as the Batchelor Flow (Batchelor, 1967). I then discuss the
conversion between a 3 dimensional flow field $\vec{v}(\vec{r})$ and the line-
of-sight component (radial velocity) available in astronomical Doppler
measurements. I describe how spectroscopic observations of spatially-varying
Doppler shifts in an astronomical spectroscopic mapping experiment might be
used to estimate the vorticity. Since this is an observational proxy for an
unmeasureable physical quantity, I call this quantity the pseudovorticity.
Section 4 presents similar analysis of a flow field chosen by construction to
have $\vec{\omega}=0$ everywhere. My conclusion is that a similar
pseudovorticity would be inferred for both flow fields, although the true
vorticity is entirely different. Section 5 points out that these conclusions
could have been anticipated from clear insights on the nature of Galactic
hydrogen flows presented by W.B. Burton in a set of articles approximately 50
years ago. Section 6 summarizes and concludes.

##  2 2\. The Measurement of Vorticity in Laboratory Turbulence

An interesting aspect of the study of turbulence is its importance, not only
to fundamental physics, but also to research in the engineering community.
Many of the most insightful books and journal articles have been written by
engineers. Given that this phenomenon has attracted the scrutiny of flinty-
eyed engineers, one would expect more solid evidence on fundamental questions
than typically can be provided by physicists, let alone astronomers. One would
expect the existence and dynamical importance of vorticity in hydrodynamic
turbulence to be well-established experimentally. Curiously, this seems not to
be the case.

Measurement of turbulent fluid motions, including vorticity, are made
utilizing a number of experimental techniques, such as hot-wire anemometry
(Hubbard, 1957; Comte-Bellot, 1976; Wallace & Foss, 1995), optical tracking of
a passive tracer (Wallace, 2009; Wallace & Vukoslavčević, 2010), and particle
image velocimetry (Wallace, 2009; Wallace & Vukoslavčević, 2010). Perhaps the
most standard method is that of hot-wire-anemometry, which consists of
measuring the cooling rate of a heated wire in a moving fluid. An empirical,
though theoretically-based expression for the response of a hot-wire
anemometer is (Hubbard, 1957)

$\displaystyle\frac{I^{2}R\alpha}{a}=C_{1}+C_{2}\sqrt{U}$ (3) $\displaystyle
a\equiv\frac{R-R_{0}}{R_{0}}$ (4)

where $I$ is the current flowing through the circuit, $U$ is the component of
the flow velocity perpendicular to the wire, $R$ is the actual resistance of
the wire, $R_{0}$ is a reference value for the resistance of the wire (defined
as the resistance when the resistor has the temperature of the fluid), and
$\alpha$ is the thermal conductivity of the fluid. The coefficients $C_{1}$
and $C_{2}$ are constants characteristic of the wire, such as platinum, that
is employed in the device. Measurement of $U$ results from measuring the
current $I$ necessary to maintain the resistance of the wire at $R$.

A hot wire anemometer is not a vector instrument. The cooling rate, and thus
the resistance $R$ depend on the angle between the wire and the flow
direction. An array of wires oriented at different angles can deduce the
vector flow velocity at a point. An array of at least 9 hot-wire anemometers,
spatially-separated and oriented at specifically chosen directions, can, in
principle, measure the vorticity at a point in the fluid. Such a hot-wire
anemometer array is called a “Kovasznay probe” after its inventor, and is
sometimes referred to as a “vorticity meter”.

Nonetheless, technical problems limit the capability of measuring vorticity
with hot-wire anemometer arrays. The size of the individual elements (wires)
is problematic. A wire which is too big will average over important spatial
scales, while a wire which is too small will have significant noise error. In
addition, the structures which hold the wires in place within the flow will
modify the flow field from what it would be in the absence of those
structures. Discussions of these issues may be found in Comte-Bellot (1976),
Wallace & Foss (1995); Wallace (2009); Wallace & Vukoslavčević (2010).

Studies of turbulence with hot-wire anemometer arrays have yielded some
results which are broadly consistent with the presence of vorticies. Kuo &
Corrsin (1972) found that the regions of fine-scale fluid fluctuations tended
to be in geometric structures which were “rods”, subsequently interpreted as
“tube-shaped regions of concentrated high vorticity” (Johnson & Meneveau,
2016).

A recent, and extremely clever technique was described by Wu, Xu, and
Bodenschatz (2019). Tiny, micron sized spheres are fabricated from a material
with nearly the same index of refraction as water, and possessing embedded
mirrors. A number of these spheres are then seeded into a turbulent flow of
water. The vorticity equals twice the angular velocity at a point. Scattering
of a light ray from a laser, and off the spheres then can measure the vector
angular velocity, and thus the vorticity, at a point in the fluid. Wu, Xu, and
Bodenschatz (2019) verify the technique by showing that it yields the
theoretically-expected distribution of vorticity in a Couette flow, but to
date the technique does not seem to have been used to obtain results in large-
inertial subrange turbulence.

The incomplete state of knowledge of vorticity in laboratory turbulence is
conveyed by a statement in the introduction of Wu, Xu, and Bodenschatz (2019):
“The vorticity field underlies the dynamics of turbulent flows. Measuring
vorticity experimentally, however, is notoriously difficult”.

In concluding this brief and incomplete sampling of the literature on
laboratory measurements of turbulence, I would emphasize the reason for
including this material at all. If, as appears to be the case, it is difficult
to clearly measure and prove the presence of vorticity in laboratory flows,
where there is control over the flow conditions and geometry, one can
reasonably expect a much more difficult problem in the case of astronomical
fluids, for which incomplete, remote sensing measurements are the only ones
available.

##  3 3\. The Batchelor Flow: A Model Flow with High Vorticity

In this section, I consider a flow field known to possess vorticity as well as
vortex stretching, so the vorticity increases with time.

Starting with the Navier-Stokes equations, the following equation for the
vorticity can be derived (Bradshaw, 1978)

$\frac{\partial\vec{\omega}}{\partial
t}+(\vec{v}\cdot\nabla)\vec{\omega}=(\vec{\omega}\cdot\nabla)\vec{v}+\mbox{
viscous terms }$ (5)

The terms on the left hand side constitute the familiar convective derivative,
while the term on the right hand side describes an “amplification” of
vorticity due to the strain field. The omitted viscous terms (Bradshaw, 1978)
describe spatial diffusion of vorticity due to viscosity. This equation shows
the basic physics content of vortex amplification by stretching; when there is
a positive derivative of the component of the flow velocity in the direction
of the vorticity, the vorticity is amplified. A clear discussion is in
Bradshaw (1978).

Since a vortex consists of “swirling” motion, for the following discussion I
utilize cylindrical coordinates ($R$, $\phi$, $z$), and take the direction of
the vorticity to be in the z direction.

###  3.1 3.1 The Batchelor Flow

Batchelor (1967) considered a flow field which would lead to vortex
stretching, and a vorticity field given by

$\vec{\omega}=0\hat{e}_{R}+0\hat{e}_{\phi}+\omega(R,t)\hat{e}_{z}$ (6)

The conditions for the vorticity to be of this form are

$\frac{\partial v_{\phi}}{\partial z}=\frac{\partial v_{R}}{\partial
z}=\frac{\partial v_{z}}{\partial R}=0$ (7)

The background flow field chosen by Batchelor (1967), which satisfies the
above relations, is

$\displaystyle v_{z}=\alpha z$ (8) $\displaystyle
v_{R}=-\left(\frac{\alpha}{2}\right)R$ (9)

where $\alpha$ is a constant with dimensions of $t^{-1}$. The azimuthal
component of the velocity, $v_{\phi}$ is contained in the vortex being
amplified. Substitution of this background flow into Equation (5) yields
Equation 5.2.9 of Batchelor (1967), an expression for the scalar vorticity
$\omega$

$\frac{\partial\omega}{\partial
t}=\left(\frac{\alpha}{2}\right)\frac{1}{R}\frac{\partial}{\partial
R}\left(R^{2}\omega\right)$ (10)

Equation (10) omits a term describing viscous diffusion of vorticity.

###  3.2 3.2 A Specific Solution

Given Equation (10), I am looking for a solution that will provide a specific
form of $\omega(R,t)$, that can then be converted to a form for
$v_{\phi}(R,t)$. I choose a separable form,

$\displaystyle\omega(R,t)=A(t)w(x)$ (11) $\displaystyle x\equiv\frac{R}{a(t)}$
(12)

where $a(t)$ is a time-dependent scale factor for the vortical flow. This form
has the Kelvin Circulation Theorem, with conservation of the circulation, in
mind from the start. Substituting this trial function $\omega(R,t)$ into
Equation (10), the following form emerges

$\dot{A}w-Aw^{{}^{\prime}}R\left(\frac{\dot{a}}{a^{2}}\right)=\left(\frac{\alpha}{2}\right)A\left[2w+\left(\frac{R}{a}\right)w^{{}^{\prime}}\right]$
(13)

where $\dot{A}=\frac{dA}{dt}$ and $w^{{}^{\prime}}=\frac{dw}{dx}$. Dividing
through by $Aw$ gives

$\frac{\dot{A}}{A}-\left(\frac{w^{{}^{\prime}}}{w}\right)R\left[\frac{\dot{a}}{a^{2}}+\left(\frac{\alpha}{2}\right)\frac{1}{a}\right]=\alpha$
(14)

Any solution to this equation for the functions $A(t)$ and $w(x)$ is a
candidate physical solution, if it behaves in a physical way and can match
appropriate initial conditions.

One obvious solution is to set the term in square brackets to zero, i.e.

$\left[\frac{\dot{a}}{a^{2}}+\left(\frac{\alpha}{2}\right)\frac{1}{a}\right]=0$
(15)

giving the solution for $a(t)$

$a(t)=a_{0}e^{-\alpha t/2}$ (16)

corresponding to a vortex which contracts with time. If this solution is used,
the resulting expression for $A(t)$ is

$A(t)=A_{0}e^{\alpha t}$ (17)

corresponding to a strengthening of the vortex as it contracts. Equations (16)
and (17) suggest the introduction of a dimensionless time $\tau\equiv\alpha
t$. Given these restrictions on the strength and scale of the vortex as it is
amplified, any form for the function $w(x)$, which describes the spatial
distribution of the vorticity, is acceptable. A physically plausible choice is
one which has a maximum on the axis of the vortex, and decreases with
increasing radial distance from the axis. The obvious choice is a Gaussian,

$w(x)=e^{-x^{2}}$ (18)

with dimensional factors collected into the normalization constant $A_{0}$.
Equations (16), (17), and (18) can be used to constitute an expression for the
time and space-dependent vorticity,

$\omega(R,t)=\omega_{0}e^{\tau}\exp\left[-\left(\frac{R}{a_{0}}\right)^{2}e^{\tau}\right]$
(19)

where $\omega_{0}$ is the initial vorticity on axis, and $a_{0}$ is the
initial radius of the vortex (defined in Equation (16)). This solution will be
used for the vortical flow in the remainder of this investigation.

Given the above expressions for the vorticity $\vec{\omega}(R,t)$ we want to
obtain an expression for the flow field in the vortex, $\vec{v}(R,t)$. From
the definition of the vorticity and Equations (6) and (7) we have

$\vec{\omega}=\omega(R,t)\hat{e}_{z}=\frac{1}{R}\frac{\partial}{\partial
R}\left(Rv_{\phi}\right)\hat{e}_{z}$ (20)

The vortical flow, $v_{\phi}(R,t)$ is azimuthally symmetric and invariant in
z.

If Equation (11) is substituted into Equation (20), and integrated from $0$ to
$R$, the following expression for $v_{\phi}$ results

$v_{\phi}(R,t)=\left[\frac{\omega_{0}a_{0}^{2}}{2}\right]\left(\frac{1}{R}\right)\left[1-e^{-R^{2}/a^{2}}\right]$
(21)

where the constant $A_{0}$ in Equation (17) is identified with the initial
vorticity $\omega_{0}$. The Kelvin Circulation Theorem is expressed by the
fact that $\omega_{0}a_{0}^{2}$ is a constant.

The solution given by Equation (21) has the property that for $R\gg a$,
$v_{\phi}\propto R^{-1}$, which is correspondent to the Biot-Savart Law in
electricity and magnetism. In this part of space, the vorticity is
approximately zero, supporting the physical relevance of the solution adopted
in Equation (15).

In comparing the flow field Equation (21) to synthetic observations, it will
be helpful to have fixed, rather than time variable, dimensionless
coordinates. I therefore pick a new dimensionless variable y,

$y\equiv\frac{R}{a_{0}}$ (22)

The flow field is then given by

$v_{\phi}(y,t)=\left[\frac{\omega_{0}a_{0}}{2}\right]\left(\frac{1}{y}\right)\left[1-\exp{(-y^{2}e^{\tau})}\right]$
(23)

where $\tau\equiv\alpha t$ is the dimensionless “age” of the stretched vortex.
I also define a new constant $V_{0}\equiv\frac{\omega_{0}a_{0}}{2}$ as the
velocity scale associated with the vortex. The expression in the last term on
the right hand side of Equation (23) shows the profile of vorticity in the
stretched and contracting vortex. Assumed expressions for $w(x)$ different
from that adopted in Equation (18) would presumably differ in the expression
in this term.

The velocity profile corresponding to Equation (23) is shown in Figure 1 for
three values of the dimensionless time $\tau$.

Figure 1: The velocity profile $v_{\phi}$ as a function of dimensionless
distance $y$ from the vortex axis for three different dimensionless times
$\tau=\alpha t=0,0.1,0.3$ (blue, yellow, green, respectively). The azimuthal
velocity reaches a peak for $y\sim 1$, then asymptotes to a irrotational,
$R^{-1}$ form for sufficiently large values of y. This plot shows that with
increasing time $\tau$, the vortex contracts and the azimuthal velocity
increases.

###  3.3 3.3 The Optically Thin Spectrum from a Batchelor Flow

In most cases, astronomical anemometry consists of the measurement of a
Doppler-shifted spectral line at a given position on the sky. The spectrum is
formed, in the optically-thin case, by contributions of emitting elements all
along the line of sight. Velocity measurements therefore consist of a weighted
average velocity along a line segment from the observer to infinity in one
direction. This is a dramatic contrast with laboratory measurements which seek
to measure the fluid velocity $\vec{v}(\vec{r})$ at a given point $\vec{r}$.
Attempts to determine spatial functions of the velocity, such as the
vorticity, therefore require consideration of the velocity profile along the
line of sight, if such is knowable. In this section, I consider the “mapping”
between the velocity field and the observed spectrum for the case of the
Batchelor Flow. In Section 4, I do the same for model flow fields chosen to be
irrotational.

A number of drastic approximations are made to simplify the analysis. These
could be relaxed in future investigations, but my reason for making them is
that with these approximations, the vortical signature should be at a maximum.
Relaxation of these restrictions, and consideration of a flow field possessing
a number of vortices (the obvious realistic case), would presumably blur the
signature of vorticity, although this assertion should be verified.

The assumptions and restrictions employed are as follows.

  1. 1.

I assume that the line radiation emitted by the fluid in the vortex is
optically-thin. This permits the intensity of the radiation to be expressed as
a line integral along the line-of-sight. In astrophysical applications, this
is often the case, but not always.

  2. 2.

I assume that line-of-sight lies entirely in a plane with a constant value of
the cylindrical coordinate $z$. This means that the axis of the vortex is
perpendicular to the line-of-sight. In this plane, we introduce a Cartesian
coordinate system $(s,t)$ (t not to be confused with the time), where $t$ is a
coordinate along the line-of-sight, and $s$ is a coordinate in the plane of
the sky, and perpendicular to the axis of the vortex. This coordinate system
is illustrated in Figure 2. This assumption clearly simplifies the analysis,
since in general, there will be two angles describing the orientation of the
vortex axis with respect to the plane of the sky. The restriction chosen
should maximize the “signal” due to the vortex.

  3. 3.

I also assume that the velocity field measured in the spectrum is exclusively
due to the velocity field in the vortex (Equation (23)), i.e. the vortex
dominates over the background fluid flow ( Equation (8) and (9)). The $z$
component of the background flow will not contribute because of the
restriction # 2 above, but $v_{R}$ will, in general, contribute to the
spectrum. My assumption will be valid if the background, “stretching” flow is
slow compared with the flow in the vortex. This assumption will always be true
if the vortex has grown sufficiently.

  4. 4.

The finite extent of the emitting region, taken to be larger than the size of
the vortex, is parameterized by a length scale $L$ in the $t$ direction. The
density of the emitting atoms or molecules is taken to be uniform for
$-L/2\leq t\leq L/2$ and zero outside. The constant density is compatible with
incompressible flow.

  5. 5.

Given the coordinate system defined in item # 2 and illustrated in Figure 2,
the Doppler shift of a fluid element depends only on the velocity component
$v_{t}$, will vary with the $t$ coordinate, and be symmetric about $t=0$. The
spectrum will be determined by the value of $s$, and change with the value of
$s$.

Figure 2: Diagram showing line-of-sight (LOS) relative to the coordinate
system used to describe the vortex. The s and t coordinates are
(dimensionless) Cartesian coordinates in the $z=0$ plane of the vortex. The
vertical arrow represents the LOS, which has all values of $t$ and a constant
value of $s$. The blue circle represents the cylindrical coordinate system
used to derive the expression for the flow in the Batchelor Vortex, Equation
(23). This circle may be considered the locus of points with constant
vorticity.

Given these assumptions and simplifications, the Doppler shift of a given
fluid element is given by

$v_{t}(t(R,\phi))=v_{\phi}(R)\cos\phi$ (24)

where $v_{\phi}(R)$ is given by Equation (21), and the cylindrical coordinate
$\phi=0$ along the positive $s$ axis. The spectrum at a given position $s$
contains a range of velocities from $v_{t}(\mbox{max})\equiv v_{\phi}(R=s)$ to
$v_{t}(\mbox{min})\equiv v_{\phi}(R=R_{\mbox{max}})\cos\phi_{\mbox{max}}$
where $R_{\mbox{max}}$ and $\phi_{\mbox{max}}$ are the values of the
cylindrical coordinates at the point where the line-of-sight enters or exits
the emitting region.

Using the definition of these parameters, one can convert from
$v_{t}(t(R,\phi))\rightarrow v_{t}(s,t)$. I also introduce dimensionless
versions of the Cartesian coordinates, choosing the initial vortex radius
$a_{0}$ as the dimensional scale,

$s=a_{0}s^{{}^{\prime}},t=a_{0}t^{{}^{\prime}},L=a_{0}L^{{}^{\prime}}$ (25)

Substituting these expressions into Equation (23), and from now on noting the
dimensionless coordinates by $(s,t)$ (i.e. dropping the primes), I have

$v_{t}(s,t)=V_{0}\left[\frac{s}{s^{2}+t^{2}}\right]\left(1-\exp[-(s^{2}+t^{2})e^{\tau}]\right)$
(26)

for $-L^{{}^{\prime}}/2\leq t\leq L^{{}^{\prime}}/2$, and again,
$V_{0}\equiv\frac{\omega_{0}a_{0}}{2}$. So,

$v_{t}=v_{t}(V_{0},L^{{}^{\prime}},\tau,s,t)$ (27)

A plot of $v_{t}$ for the Batchelor Flow (Eq. 25) is shown in Figure 3. Curves
are shown for three values of the dimensionless “transverse” coordinate $s$.

Figure 3: The line-of-sight component of the fluid velocity ($v_{t}$) in the
vortex as a function of $t$, the coordinate along the line of sight. For all
three curves, the normalization constant of the velocity $V_{0}=1.0$, the
dimensionaless time $\tau=0$, and the dimensionless thickness of the emitting
region $L=10$. The three curves give $v_{t}$ for spatial offset values
(distance from the axis of the vortex, dimensionless) of $s=0.25$ (blue),
$s=0.50$ (yellow), and $s=1.00$ (green). In all cases, it is seen that the
line-of-sight component of the velocity maximizes at $t=0$, and diminishes to
a low value near the boundary of the emitting region.

It may be seen that for all three values of $s$, the maximum value of $v_{t}$
occurs at $t=0$. This figure also shows that for each velocity within the
spectrum, there are two points along the line of sight $\pm t$ with that
value. For sufficiently large values of $t\rightarrow L^{{}^{\prime}}/2$, the
velocity asymptotes to zero. Figure 3 also shows that the LOS component of the
velocity, $v_{t}$, ranges from a maximum value of
$v_{t}(\mbox{max})=v_{t}(s,0)$ to a minimum value
$v_{t}(\mbox{min})=v_{t}(s,\pm L/2)$.

###  3.4 3.4 Relating the Velocity Field to the Emission Spectrum

With a radio or optical telescope, one measures the intensity (or flux density
if the source is unresolved) of emission as a function of frequency. It is
conventional in astronomy to change the frequency variable from the
fundamental wavefrequency $f$ to the associated Doppler shift velocity $V$.

In astronomical sources, the LOS component of the fluid velocity will change
with position along the line of sight, leading to a broadened spectrum. In
section 3.3 above, I gave an expression (Equation 26) for the LOS component of
the vortex velocity of a Batchelor Flow as a function of the coordinates
$(s,t)$. In the present section, I use this expression to obtain an equation
for the (potentially observable) spectrum of a Batchelor vortex. In the
optically-thin case, the incremental change in the intensity as a function of
Doppler velocity $V$ with LOS position $t$ is333For clarity of presentation, I
use the variable $V$ to indicate a Doppler-shifted velocity as would be
measured at a telescope, to distinguish it from $v_{t}$, which is the LOS
component of the fluid velocity at a given position in the fluid.

$dI(V)=j(V,t)a_{0}dt$ (28)

where again, $t$ is a dimensionless coordinate (Figure 2), and $j(V,t)$ is the
emission coefficient.

It is reasonable to assume that the intrinsic line width of the radiation is
much smaller than the spectral broadening due to velocity variations along the
LOS. If the gas properties are otherwise uniform in the whole region for
$-L/2\leq t\leq L/2$, then changes in the emission coefficient are due to
changes in the Doppler velocity $V$ at which the emission occurs, so

$j(V,t)=\frac{J_{0}}{a_{0}}\delta(V-v_{t}(t))$ (29)

where the normalization coefficient $J_{0}$ (a measure of the absolute
intensity of the radiation) accommodates the dimensional factors needed in the
use of Dirac delta functions. Thermal broadening, of course, will generally be
larger than the intrinsic line width, and could be comparable to that of the
turbulent flow velocities. However, thermal broadening can be accommodated by
convolving a Gaussian function representing that broadening with the line
profiles calculated here. Equation (27) then becomes

$\displaystyle dI(V)=J_{0}\delta(V-v_{t}(t))dt=J_{0}\delta(f(t))dt$ (30)
$\displaystyle f(t)\equiv V-v_{t}(t)$ (31)

To simplify the expression $\delta(f(t))$, I use the following identity from
the algebra of delta functions (Dennery and Krzywicki, 1967)

$\delta(f(t))=\sum_{i}^{N}\frac{1}{|\frac{df}{dt}|_{t=t_{i}}|}\delta(t-t_{i})$
(32)

where $t_{i}$ is the $i^{th}$ root of $f(t)=0$. In the case of the Batchelor
Vortex, there will be two roots at $\pm t$ and the same value of
$|\frac{df}{dt}|$. Substitution of Equation (32) into Equation (30) and
integration over $-L/2\leq t\leq L/2$ yields the expression for the “observed”
spectrum of the Batchelor Vortex

$\displaystyle
I(V)=2J_{0}\left(\frac{1}{|-\frac{dv_{t}}{dt}|_{t=t_{0}}|}\right)\mbox{ if
}v_{t}=V\mbox{ at any point where }t=t_{0}$ (33) $\displaystyle=0\mbox{ if
}v_{t}\neq V\mbox{ at any point on LOS}$ (34)

Equation (33) incorporates the fact that, for the Batchelor Vortex, there will
be two points along the LOS at $t=\pm t_{0}$, if there is any point at which
$f(t)=0,V=v_{t},\mbox{ for }-L/2\leq t\leq L/2$.

For the case of the flow field in the Batchelor Vortex, Equation (26), we have

$\frac{dv_{t}}{dt}=V_{0}\left[-\frac{2st}{(s^{2}+t^{2})^{2}}\left(1-\exp(-(s^{2}+t^{2})e^{\tau})\right)+\frac{s}{s^{2}+t^{2}}\left(2te^{\tau}\exp(-(s^{2}+t^{2})e^{\tau})\right)\right]$
(35)

The dimensional constant $V_{0}$ is an overall scale factor that sets the
velocity scale of the vortex. The function Equation (35), substituted into
Equation (33), gives the expression for the spectrum.

With the above equations, the emission spectrum of a Batchelor Vortex can be
calculated. This spectrum depends on the velocity scale of the vortex,
$V_{0}$, the dimensionless age $\tau$, and the scale size of the vortex
$a_{0}$. In what follows, I take $\tau=0$; converting to times $\tau\neq 0$
presents no difficulties. I set $V_{0}=1$, meaning any vortical flow can be
studied by multiplying by the appropriate velocity scale. Finally, I leave the
distances $s$ and $t$ dimensionless so they, also, can be scaled to any flow
of interest.

The spectrum of a vortex is calculated using Equations (26), (33), and (35).
The specific steps are as follows.

  1. 1.

A velocity model is chosen, using Equation (26), with parameters $V_{0}=1$ and
$\tau=0$ for the results presented here.

  2. 2.

The range of Doppler velocities of interest are chosen, ranging from the
maximum value $v_{t}(\mbox{max})$ to $v_{t}(\mbox{min})$ (defined in Section
3.3 following Equation (24)).

  3. 3.

A set of values of $V$ are chosen $v_{t}(\mbox{min})\leq V\leq
v_{t}(\mbox{max})$. These correspond to the “channels” in an observed
spectrum. They are chosen to be at equal spacing in $V$ between the limits. In
a typical calculation carried out, the number of channels $N$ was arbitrary
chosen to be 31 - 32. Any value of $N$ could be chosen.

  4. 4.

The derivative of the LOS component of the velocity, $\frac{dv_{t}}{dt}$ is
calculated using Equation (35).

  5. 5.

For each of the $N$ “channel velocities” $V_{i},i=1,N$, the corresponding
value of $t_{i}$ is obtained by solving the equation

$v_{t}(t_{i}(V_{i}))-V_{i}=0$ (36)

for $t_{i}$. The resulting solution $t_{i}$ gives the position along the LOS
at which $v_{t}=V_{i}$, the position at which Doppler-shifted radiation would
be emitted at the observed velocity $V_{i}$

  6. 6.

Using this set of $t_{i}$ values, the value of
$|-\frac{dv_{t}}{dt}|_{t=t_{i}(V_{i})}|$ for each of the channels is
calculated, and used in Equations (32) and (33) to calculate the “observed”
spectrum. This spectrum is the final output of the analysis.

An illustration of this calculation is shown in Figure 4. Spectra are shown
for two lines of sight with $s=0.5,1.0$, i.e. through the central part of the
vortex.

Figure 4: The spectrum as given by Equation (33), in units of $2J_{0}$, for
the case of $\tau=0$ and $V_{0}=1$. The spectrum in gold squares represents
the spectrum observed at a dimensionless offset distance from the vortex axis
$s=0.5$. The spectrum with blue solid circles is the spectrum for $s=1.0$.
Both lines of sight pass within the vortex, and thus regions of high
vorticity. The dashed vertical line indicates the maximum Doppler shift
velocity for the LOS with $s=0.5$, and the dot-dash vertical line is the
limiting velocity for the case $s=1.0$. Both spectra show a peak at zero
Doppler shift, corresponding to gas outside the vortex, or vortical flow
perpendicular to the LOS, a minimum in the emission at intermediate
velocities, and a peak at the limiting velocity corresponding to the peak LOS
component of the vortical flow. The detailed characteristics of the spectrum,
most importantly the peak at the limiting velocity, change with the offset
distance $s$.

The spectra in both cases show common features; strong emission at low Doppler
velocities corresponding to emission from gas that is stationary or slowly
moving with respect to the observer, weaker emission at intermediate
velocities corresponding to more pronounced motions associated with the
vortex, and finally a rise to a “spectral peak” at the maximum Doppler
velocity $v_{t}(\mbox{max})$. There is no emission for velocities higher than
this value. These features can be understood on the basis of the inverse
dependence of the intensity on the derivative of $v_{t}$ (Equation 33), and
the form of this function as shown in Figure 3.

The details of the spectrum, most importantly the Doppler velocity of the peak
associated with $v_{t}(\mbox{max})$, depend on the properties of the vortex
and the value of the offset distance $s$. The spectrum in Figure 4 is for
$s\geq 0$, i.e. lines of sight on the side of the vortex axis corresponding to
$v_{t}\geq 0$. As may be seen in Figure 2, for lines of sight at negative $s$,
corresponding to the other side of the vortex, the spectrum would be
antisymmetric, the same in form but for Doppler velocities which are negative
(blueshifted). This analysis thus suggests a possibly simple, qualitative
observational test for vortices in astrophysical flows. Spectral mapping of
the vicinity of a vortex might yield a characteristic pattern, qualitatively
similar to spectral observations of spiral galaxies (see Section 5 below).

For lines of sight which pass well outside the core of the vortex, Doppler-
shifted gas will appear in the spectrum, but the vorticity along the LOS will
be close to zero (Equation 18). Figure 5 shows spectra calculated in the same
way, and presented in the same format, as Figure 4. The two spectra correspond
to offset (from the vortex axis) distances of 3 and 4, as described in the
figure caption. It is clear that the same qualitative features are present in
these spectra as are discussed above for lines of sight through the vortex.
This indicates that a quantitative, rather than purely qualitative analysis is
necessary to test for the presence of vorticity in emission spectra.

Figure 5:  Spectra, in the same format as Figure 4, for line of sight
distances of $s=3$ (gold squares) and $s=4$ (blue circles), outside the vortex
where the flow is largely irrotational. The spectra are qualitatively the same
as those measured through the core of the vortex, although the quantitative
details differ. Vertical lines again indicate the limiting velocities for each
spectrum.

###  3.5 3.5 Determination of the “Pseudovorticity” from the Spectrum

To this point, I have discussed the emission spectrum from a vortex of
specified properties. and noted the characteristics of that spectrum. In this
section, I consider the astronomically-relevant question of taking a spectrum,
which presumably roughly resembles the idealized spectra in Figures 4 and 5,
and inferring the vorticity.

The prime observational feature in such a spectrum is the “spectral line” at
the maximum vortex velocity $v_{t}(\mbox{max})=v_{t}(s,0)$. As indicated in
this expression, this peak is a function of the (dimensionless) coordinate
$s$.

The plane containing the line of sight is, by construction, one with Cartesian
coordinates $(x,y)\rightarrow(s,t)$. Retaining the $(x,y)$ coordinates, the
$z$ component of the vorticity, which is the only one which is nonzero, is

$\omega_{z}=\left(\frac{\partial v_{y}}{\partial x}-\frac{\partial
v_{x}}{\partial y}\right)$ (37)

A potentially observable change of $v_{t}(\mbox{max})$ with $s$ corresponds to
the first term on the left side of Equation (37), albeit path-integrated.
However, the second term is completely unknowable, since neither the numerator
nor the denominator can be measured.

Raymond et al (2020a, b) used a statistical argument that the variances of the
(spatially varying) terms on the RHS of Equation (37) would be equal. They
therefore multiplied their measured gradient by $\sqrt{2}$ to obtain their
estimate of the vorticity. In the present study, the vortex responsible for
the model spectrum is a deterministic flow, not a random field of vorticity,
so a statistical argument is not used. Instead, it is noted that the magnitude
of the vorticity in the flow field given by Equation (23) depends only on
$\sqrt{s^{2}+t^{2}}$, so the two terms in Equation (37) have equal magnitudes.
I there use a symmetry argument to the effect that on average,

$|\frac{\partial v_{y}}{\partial x}|\simeq|\frac{\partial v_{x}}{\partial y}|$
(38)

and that the signs of the two terms would be opposite. I therefore multiply
the observed gradient by a factor of 2.

With the above uncouth assumptions, I introduce an observational estimate of
the vorticity in an astronomical fluid, the pseudovorticity $\omega_{\psi}$,
defined as

$\omega_{\psi}\equiv 2\left(\frac{\Delta v_{t}(\mbox{max},d)}{\Delta
s_{d}}\right)=\frac{2V_{0}}{a_{0}}\left(\frac{\Delta v_{t}(\mbox{max})}{\Delta
s}\right)$ (39)

where the subscript “d” in the first terms on the left hand side indicate the
dimensional forms of the LOS velocity and transverse coordinate s, and the
second set of terms on the left hand side are the dimensionless versions shown
in Figures 4 and 5.

I now apply these notions to the spectra shown in Figures 4 and 5. Begin with
Figure 4 for the Batchelor Vortex, applying Equation (39) with values
“measured” from Figure 4. For this calculation, $V_{0}=a_{0}=1$. The change in
dimensionless coordinate $\Delta s=0.5$, and the shift in the dimensionless
maximum velocity $\Delta v_{t}(\mbox{max})=0.632-0.442=0.190$, so
$\omega_{\psi}=0.76$.

To see if this rude estimate is approximately valid, I compare it to the
“true” vorticity at dimensionless distances of 0.5 and 1.0 from the vortex
axis. Using Equation (19) for $\omega(s=0.5,t=0,\tau=0)\mbox{ and
}\omega(s=1.0,t=0,\tau=0)$, I obtain values of 1.55 and 0.74, respectively.
The conclusion for this case is that the observationally-inferred
pseudovorticity yields the true value within a factor of 2.

This exercise can be repeated for the “observations” of Figure 5, which
simulate a pair of lines of sight far from the vortex axis, where the
vorticity is close to zero. Using Equation (19) for the case shown in Figure 5
yields values of $2.4\times 10^{-4}\mbox{ and }2.2\times 10^{-7}$ for $s=3,4$
respectively. Application of Equation (39) yields $\omega_{\psi}=0.16$.

The calculated pseudovorticity is substantially higher than the true, nearly
zero value. However, it is substantially smaller than that which would be
inferred for observed lines of sight through the core of the vortex. This
suggests (to a self-deluded individual) that we could use this type of
analysis to distinguish between true high and low vorticity media, and perhaps
have a semi-quantitative measure of the vorticity in a fluid.

However, it is obvious that the low vorticity case discussed here concerns an
LOS at great distances from the vortex axis, where the azimuthal flow velocity
is also substantially lower. A smaller pseudovorticity would then
automatically follow. What is necessary to check this analysis is modeling an
irrotational flow with velocities comparable to those characteristic of the
case displayed in Figure 4. That is the subject of the next section.

##  4 4\. The Spectrum and Pseudovorticity of an Irrotational Flow

To test the reliability of the pseudovorticity as a measurable, it is
desirable to consider a flow for which the overall flow scale factor $V_{0}$
is adjustable, but which is irrotational by construction. This will be
introduced in the following subsection. I concede at the outset that this
might be an academic excercise, in that real astrophysical flows may not
conform with mathematical idealizations.

###  4.1 4.1 A Potential Flow

In this section, I consider a flow which possesses some of the features of the
strong vortex of the Batchelor Flow, but which nonetheless has zero vorticity
throughout the flow field. The way to guarantee to this is to have the flow
field defined by a potential function $\Phi$, resulting in potential flow,

$\vec{v}=\nabla\Phi$ (40)

A flow of this form guarantees $\nabla\times\vec{v}=0$.

If we further assume that the flow is incompressible, so
$\nabla\cdot\vec{v}=0$, then the potential function $\Phi(\vec{r})$ satisfies
Laplace’s equation,

$\nabla^{2}\Phi=0$ (41)

In selecting the potential function $\Phi$, I pick the same geometry as in
Section 3, and initially choose cylindrical coordinates.

This is a standard problem in undergraduate Electricity and Magnetism (e.g.
Reitz and Milford, 1967), where it is used to discuss the electric potential
and electric field around a conducting cylinder immersed in a uniform electric
field. In the present case, the problem of interest is that of an impermeable
cylinder immersed in a uniform flow. A visualization of the geometry and flow
is shown in Figure 6.

Figure 6: Figure, similar in format to Figure 2, but showing the flow field
corresponding to a potential flow. The vertical ray corresponds to an LOS with
$s=1.5$. The hue of the plotted vectors conveys the magnitude of $\vec{v}$.
The scale of the system is determined by the radius of the effective cylinder,
$R_{0}$, which is taken to be 1 in this figure.

The solution to Laplace’s Equation in cylindrical coordinates can be expressed
in a series expansion of terms characterized by a “quantum number” $n$ (Reitz
and Milford, 1967)

$\Phi(R,\phi)=A_{0}+B_{0}\ln
R+A_{1}^{c}R\cos\phi+A_{1}^{s}R\sin\phi+B_{1}^{c}\left(\frac{1}{R}\right)\cos\phi+B_{1}^{s}\left(\frac{1}{R}\right)\sin\phi+\ldots$
(42)

The solution to the problem of interest, i.e. the appropriate set of
coefficients in Equation (42), is found by incorporating the boundary
conditions. In this case, boundary conditions are

$\displaystyle\vec{v}(\vec{r}\rightarrow\infty)=+V_{0}\hat{e}_{t}$ (43)
$\displaystyle v_{n}(R=R_{0})=v_{R}(R=R_{0})=0$ (44)

Where $\hat{e}_{t}$ is the unit vector in the Cartesian $t$ coordinate
introduced in Section 3.3, and $v_{n}(R=R_{0})$ is the component of the
velocity normal to the surface of the cylinder at the surface of the cylinder.
It should be noted that the inner boundary condition, Equation (44), differs
from that used in undergraduate studies of electrostatics. It is also worth
emphasizing that the “cylinder” is a mathematical artiface to produce a flow
with roughly realistic properties, and need not correspond to an actual
astronomical object.

To make an irrotational flow field which simulates that of a vortical flow
with similar geometry to that considered in Section 3, I define $\phi=0$ to be
along the $t$ axis in the direction of $t\rightarrow-\infty$, and flowing in a
counterclockwise direction for $s>0$ (see Figure 6).

The boundary conditions (43) and (44) are satisfied if the coefficients in
Equation (42) satisfy the following:

$A_{1}^{s}=B_{0}=0;A_{1}^{c}=-V_{0};B_{1}^{s}=0;B_{1}^{c}=R_{0}^{2}A_{1}^{c}=-V_{0}R_{0}^{2}$
(45)

All higher order coefficients, e.g. $B_{n}^{s},A_{n}^{c}$, for $n>1$, are
zero.

With the boundary conditions selected, the solution for the potential function
$\Phi$ is

$\Phi(R,\phi)=-V_{0}\left[R+\left(\frac{R_{0}^{2}}{R}\right)\right]\cos\phi$
(46)

The corresponding velocity field, in cylindrical coordinates, is

$\vec{v}(R,\phi)=\hat{e}_{R}\left[-V_{0}\cos\phi+V_{0}\left(\frac{R_{0}^{2}}{R^{2}}\right)\cos\phi\right]+\hat{e}_{\phi}\left[V_{0}\sin\phi+V_{0}\left(\frac{R_{0}^{2}}{R^{2}}\right)\sin\phi\right]$
(47)

To compare this flow with that of the Batchelor Flow, I write the flow
velocity in the Cartesian coordinate system of Figure 2,

$\vec{v}(s,t)=v_{s}(s,t)\hat{e}_{s}+v_{t}(s,t)\hat{e}_{t}$ (48)

where $s,t$ are again dimensionless coordinates, de-dimensionalized like
Equation (25), except in the present case the dimensional coefficient is
$R_{0}$, the radius of the effective cylinder rather than the Gaussian radius
of the Batchelor Vortex.

The resulting components of the flow velocity are

$\displaystyle v_{s}(s,t)=-\frac{2V_{0}st}{(s^{2}+t^{2})^{2}}$ (49)
$\displaystyle
v_{t}(s,t)=V_{0}\left[1+\frac{s^{2}-t^{2}}{(s^{2}+t^{2})^{2}}\right]$ (50)

This flow field, which will be used in our analysis of the model spectrum and
deduction of the pseudovorticity for an irrotational flow field, is plotted in
Figure 6. The flow speed is higher than $V_{0}$ close to the cylinder, a
consequence of the Equation of Continuity.

The flow field in Figure 6 shows a degree of similarity with the Batchelor
Flow; for $s>0$ there is an azimuthal flow in the counterclockwise direction.
In the next subsection, I examine the degree to which the spectrum emergent
from this flow resembles that from the Batchelor Vortex discussed in Section
3.

###  4.2 4.2 Potential Flow: Spectrum and Pseudovorticity

To calculate the spectrum of radiation, and deduce the pseudovorticity for the
case of the potential flow field, I follow the same steps as in Section 3 for
the Batchelor Flow. Equation (32) is general, and is used here. That equation
shows that the observed spectrum is determined by the profile of $v_{t}(s,t)$
for fixed s. The spectrum is dominated by “emission lines” at velocities which
correspond to zeros of $\frac{dv_{t}}{dt}$.

The expression for $v_{t}(s,t)$ used here is Equation (50), and it is plotted
as a function of $t$ in Figure 7.

Figure 7: The line-of-sight component of the fluid velocity ($v_{t}$) for the
case of potential flow given by Equations (49) and (50) as a function of $t$,
the coordinate along the line of sight. For all three curves, the
normalization constant of the velocity $V_{0}=1.0$, and the dimensionless
thickness of the emitting region $L=10$. The three curves give $v_{t}$ for
spatial offset values (distance from the center of the effective “cylinder”,
dimensionless) of $s=1.5$ (blue), $s=2.0$ (yellow), and $s=3.00$ (green). As
in the case of the Batchelor Vortex shown in Figure 3, and for all three
offset distances, it is seen that the the line-of-sight component of the
velocity maximizes at $t=0$, and diminishes to a low value near the boundary
of the emitting region. However, in contrast to the Batchelor Vortex, the LOS
component $v_{t}$ has another extremum at an intermediate value of $t$.

Figure 7 shows that for $s>0$, there are three values of t for which
$\frac{dv_{t}}{dt}=0$, as opposed to two for the Batchelor Vortex (Figure 3).
These values are as follows.

  1. 1.

For $|t|\gg 1,\frac{dv_{t}}{dt}\rightarrow 0$. This corresponds to the
“unperturbed flow”.

  2. 2.

For intermediate values $2\leq t\leq 4$, $v_{t}<V_{0}$. This corresponds to
“steering” or “deflection” of the flow away from the line of sight. In this
intermediate range, there is a minimum value of $v_{t}$ (“blueshifted” with
respect to $V_{0}$) and a corresponding case of $\frac{dv_{t}}{dt}=0$. As
shown in Figure 7, the location of this minimum value of $v_{t}$ depends on
$s$.

  3. 3.

Finally, for $t\rightarrow 0$, $v_{t}$ increases to $v_{t}>V_{0}$, reaching a
maximum at $t=0$, as is the case for the Batchelor Vortex. This maximum is a
consequence of the equation of continuity as the flow is blocked by the
cylinder, although I again emphasize that the literal model of the flow field
is not the point of interest here.

Figure 7 allows us to sketch a qualitative spectrum of the emission from the
potential flow, as shown in Figure 8.

Figure 8: Heuristic representation of the spectrum for the potential flow
displayed in Figure 6. The spectrum from such a flow would show three peaks: a
peak corresponding to flow unshifted with respect to the net flow at $V_{0}$
(central peak), a blueshifted peak when the flow is being diverted away from
the line of sight (smaller peak to the left), and a strong redshifted peak
when the LOS speed is higher than $V_{0}$ around $t=0$ (larger peak on right).
The form of this spectrum could be inferred from Figures 6 and 7 and Burton’s
insights on Galactic structure from 21 cm spectra (see Section 5).

The large central peak (corresponding to $V=V_{0}=5$) is undeflected and
unperturbed gas. The dominance of this feature is determined by the size scale
of the total emitting region ($L$) relative to the size scale of the flow
perturbation ($R_{0}$). The peak at lower velocities ($V\simeq 2$) corresponds
to flow that is “deflected but not accelerated”. The absence of emission at
velocities lower than this peak ($V<V_{min}$) is complete. Finally, the peak
with $V=V_{max}>V_{0}$ is dominated by emission at $t\simeq 0$. Again, there
is no emission at $V>V_{max}$.

As indicated in Equations (33) and (34), the spectrum is determined by the
function $\frac{dv_{t}}{dt}$. For the potential flow field (47) - (50), this
function is

$\frac{dv_{t}}{dt}=\left(\frac{V_{0}}{R_{0}}\right)\frac{-2t}{(s^{2}+t^{2})^{2}}\left[1+\frac{2(s^{2}-t^{2})}{(s^{2}+t^{2})}\right]$
(51)

The leading term on the right hand side makes the overall expression
dimensionally correct. The variables $s,t$ in Equation (51) are dimensionless,
as in Section 3.

In what follows, I carry out the analysis for the “redshifted” feature of the
generic spectrum of Figure 8, corresponding to the flow around $t\sim 0$. This
arises from a part of the flow which crudely resembles the Batchelor Vortex in
the sense of consisting of an azimuthal flow. Emission of this feature will be
present in the velocity range $V_{0}\leq V\leq V_{max}$. As in the case of the
Batchelor Vortex, two values of $t$ contribute to each value of $V$ in this
range.

Spectra for the potential flow field, calculated with Equations (33) and (34)
are shown in Figure 9, for two values of s.

Figure 9:  Spectra, in the same format as Figures 4 and 5, for the case of the
potential flow discussed in Section 4.1. Different colored symbols correspond
to line of sight distances of $s=1.5$ (gold squares) and $s=2$ (blue filled
circles). Vertical lines again indicate the limiting velocities for each
spectrum. The spectra show the same qualitative features as those of the
Batchelor Vortex (Figure 4), although the vorticity is identically zero in the
case shown here.

What is immediately obvious from Figure 9 is that the spectra have the same
qualitative form as Figure 4 for the Batchelor Vortex. This similarity is a
consequence of Eq (33,34) and that fact that, for a “circulating flow”, there
is a point in the flow for which $\frac{dv_{t}}{dt}=0$. The principal result
of Figure 9 is that this flow field, like the Batchelor Vortex, produces a
spectrum with a peak at low velocities ($V=V_{0}$), a fall-off to a minimum
intensity at intermediate velocities, then a rise to a “spectral line” at
$V=V_{max}$.

I now proceed to calculate the pseudovorticity in the same way as in Section
3, utilizing Equation (39). Once again, for the spectra in Figure 9, the
dimensional factors $V_{0},R_{0}$ are taken to be unity. For the spectra in
Figure 9, $\Delta v_{t}(max)=0.19$ and $\Delta s=0.50$, yielding a
pseudovorticity $\omega_{\psi}=0.76$. This is exactly the pseudovorticity
inferred from the spectrum produced by the Batchelor Vortex, even though in
the former case the flow field consisted of an idealized vortex, and in the
present case the flow field is mathematically constructed to possess zero
vorticity.

An extreme statement of this result is that a path-integrated spectrum,
providing information on only one component of a three-dimensional flow field,
does not preserve or convey information on the vorticity in the fluid. Insight
into this result is given in the next section. Before leaving this section, it
should be pointed out that spectroscopic mapping observations could, in
principle, distinguish between the idealized flows considered here. As noted
in Section 3.4, in the case of the Batchelor Vortex (or any other simple
vortical flow), the Doppler shifts for $s<0$ would be equal in magnitude and
opposite in sign to those for $s>0$. In the case of the potential flow
considered immediately above, spectra at $\pm s$ would be the same (Doppler
shifts equal in magnitude and sign).

##  5 5\. Vorticity Measurements from Spectroscopy and Burton’s Admonitions on
Galactic Structure from Spectroscopy

“There is great unity in physics” 444The first sentence in the document for a
laboratory exercise I did while an undergraduate. I forget what the experiment
was, but remember the inspiring invocation.

The conclusion of the paper to this point is that spatially-resolved spectra
cannot yield a certain measurement of the vorticity in a fluid. While a test
case in which vorticity is present, the Batchelor Vortex, yields satisfactory
results, a velocity field constructed to have zero vorticity appears to have
similar vorticity. I have claimed that any “circulating” flow, that is one
with a substantial azimuthal flow over part of the fluid, would yield the
observational signature of vorticity shown in Figures 4, 5, and 9.

This result could actually have been anticipated from work in a very different
area of astronomy, the study of the distribution of hydrogen gas in the Galaxy
from observations of the 21 cm line. This similarity motivated the statement
about the unity of physics given at the start of this section. In the early
and mid 1970s, Butler Burton of the National Radio Astronomy Observatory
pointed out that integrated (through a long line of sight through the Galaxy)
21 cm spectra could give the impression of spatially-localized regions of
hydrogen, even when the hydrogen gas was uniformly distributed. Burton pointed
out that, even if the gas were uniformly distributed along the line of sight
rather than clumped in clouds or spiral arms, bright emission at a specific
velocity would be observed if there were small changes in the line-of-sight
velocity for large distances along the LOS (Burton, 1971). Fainter emission
would be measured at other velocities. This situation could arise in a number
of ways, but one of the more obvious was for the gas interior to the orbit of
the Sun being in circular motion. Burton pointed out that the gradient in LOS
velocity with distance along the LOS was small in the vicinity of the tangent
point. Burton (1971) succinctly stated the effect as “…for any reasonable
rotation law, the velocity observed from regions near the subcentral point
changes very slowly along the line of sight, so that the profile contains a
contribution from a long pathlength near the maximum velocity. This effect
results in the high-velocity ridge pattern characteristic of both the observed
and the model contour maps.”. Another comment in this paper is that “
…intensities at velocities at which $|\frac{dV}{dr}|$ (the equivalent of
$\frac{dv_{t}}{dt}$ in the present paper) is small will be enhanced on the
profiles”.

This effect is also mentioned in the magisterial review of Galactic HI
observations and interpretation by Dickey and Lockman (Dickey and Lockman,
1990), where the term “velocity crowding” is introduced, defined as “…a long
distance can be “crowded” into a small velocity interval”. The net effect is
that emission from a large distance is concentrated in a small range of
observed velocities, resulting in a prominent “spectral line” in the spectrum.

These results from studies of the Galactic-scale distribution and kinematics
of Galactic hydrogen are highly analogous to the case I have discussed for
turbulent, vortical motion in astrophysical fluids that are emitting
optically-thin line emission. Those who wish to carry out turbulence
measurements from spectroscopy must be aware of the admonition in Burton
(1971), “any interpretation of the observations must deal with these
geometrical effects ”.

##  6 6\. Summary and Conclusions

  1. 1.

A vortex viewed perpendicular to the axis of vorticity will produce an
emission spectrum, in the optically-thin case, which would allow the vorticity
in the fluid to be retrieved from astronomical measurements. This statement is
predicated on the line of sight containing a single vortex which is resolved
by the astronomical observations, say the beam of a radio telescope.

  2. 2.

In this optimum case, the estimate of vorticity which can be retrieved from
the observations, with assumptions about the symmetry of the flow, is a
reasonable estimate of the true vorticity.

  3. 3.

Fluid flows which are irrotational, i.e. having zero vorticity, but possessing
azimuthal flow, or at least a radius of curvature perpendicular to the
streamlines for some of the distance along the line of sight, can also yield
an observed spectral signature that would indicate the presence of vorticity
in the radiating medium. In this case, the observations would yield a finite
value whereas the true vorticity is zero.

  4. 4.

As a result, remote sensing detections of fluid vorticity must be considered
ambiguous.

  5. 5.

It is beyond the scope of this paper to consider the obvious fact that real
observations of a turbulent fluid will generally include contributions of
numerous vortices of different magnitude, sign of vorticity, and orientation
with respect to the line of sight. One suspects that the aggregate spectrum
would contain even less information on the properties of the vorticity field
than the idealized case of a single vortex considered here.

  6. 6.

The idealized model spectra calculated here (Figures 4, 5, and 9) would
suggest that observational searches for vorticity analyse the entire line
profile (if accessible) rather than rely on measurement of line centroids. The
full spectrum might permit identification of features corresponding to
$\frac{dv_{t}}{dt}\simeq 0$.

I thank John Raymond of the Harvard-Smithsonian Center for Astrophysics for
reading the first version of this paper, and providing interesting and helpful
comments.

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