# On the turbulence driving mode of expanding HII regions

Shyam H. Menon,1 Christoph Federrath,1 Rolf Kuiper2
1Research School of Astronomy and Astrophysics, Australian National
University, Canberra, ACT 2611, Australia
2Institute of Astronomy and Astrophysics, University of Tübingen, Auf der
Morgenstelle 10, D-72076 Tübingen, Germany E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We investigate the turbulence driving mode of ionizing radiation from massive
stars on the surrounding interstellar medium (ISM). We run hydrodynamical
simulations of a turbulent cloud impinged by a plane-parallel ionization
front. We find that the ionizing radiation forms pillars of neutral gas
reminiscent of those seen in observations. We quantify the driving mode of the
turbulence in the neutral gas by calculating the driving parameter $b$, which
is characterised by the relation
$\sigma_{s}^{2}=\ln({1+b^{2}\mathcal{M}^{2}})$ between the variance of the
logarithmic density contrast $\sigma_{s}^{2}$ (where $s=\ln({\rho/\rho_{0}})$
with the gas density $\rho$ and its average $\rho_{0}$), and the turbulent
Mach number $\mathcal{M}$. Previous works have shown that $b\sim 1/3$
indicates solenoidal (divergence-free) driving and $b\sim 1$ indicates
compressive (curl-free) driving, with $b\sim 1$ producing up to ten times
higher star formation rates than $b\sim 1/3$. The time variation of $b$ in our
study allows us to infer that ionizing radiation is inherently a compressive
turbulence driving source, with a time-averaged $b\sim 0.76\pm 0.08$. We also
investigate the value of $b$ of the pillars, where star formation is expected
to occur, and find that the pillars are characterised by a natural mixture of
both solenoidal and compressive turbulent modes ($b\sim 0.4$) when they form,
and later evolve into a more compressive turbulent state with $b\sim
0.5$–$0.6$. A virial parameter analysis of the pillar regions supports this
conclusion. This indicates that ionizing radiation from massive stars may be
able to trigger star formation by producing predominately compressive
turbulent gas in the pillars.

###### keywords:

MHD – turbulence – ISM – star-formation – HII-regions

††pubyear: 2020††pagerange: On the turbulence driving mode of expanding HII
regions–10

##  1 Introduction

Stars predominantly form in dense, gravitationally-bound gas inside giant
molecular clouds (GMCs). Massive stars or clusters (OB associations) in these
clouds produce UV radiation that dissociates and photoionizes the surrounding
gas and results in the expansion of an ionization front (HII region).
Photoionization increases the thermal pressure within GMCs, has the ability to
mechanically unbind and destroy them (Williams & McKee, 1997; Matzner, 2002),
and drives turbulence that may control the formation of the next generation of
stars (Elmegreen & Scalo, 2004; Mac Low & Klessen, 2004; McKee & Ostriker,
2007; Federrath & Klessen, 2012; Padoan et al., 2014; Krumholz & Federrath,
2019).

The expanding HII regions are also known to sculpt the surrounding neutral gas
into structures reminiscent of the iconic ’Pillars of Creation’ imaged by the
Hubble Space Telescope (Hester et al., 1996). Since then, there have been a
wealth of observations using multiwavelength surveys that image these pillars
and related structures such as globules, energetic evaporating globules
(EEG’s) and proplyds, and deduce dynamical quantities in and around them
(Preibisch et al., 2012; Klaassen et al., 2014; Hartigan et al., 2015;
Schneider et al., 2016; Klaassen, P. D. et al., 2018; Klaassen et al., 2019).
Models proposed to explain their formation lie broadly in two categories: the
classic collect-and-collapse model by Elmegreen & Lada (1977) where the HII
region sweeps up and accumulates cold gas creating density enhancements and
eventually pillars in their shadows; or the more recent radiation-driven
implosion (RDI) model where clouds with pre-existing density enhancements are
sculpted to form pillars by impinging ionizing radiation. In these studies the
density enhancements are modelled as Bonnor-Ebert spheres or are seeded by
turbulence, and they naturally produce the observed morphologies and dynamics
of pillars (Mellema et al., 2006; Gritschneder et al., 2010; Mackey & Lim,
2010; Walch et al., 2012; Tremblin et al., 2013).

Direct signatures of star formation are observed at the tip of these pillars
through jets and outflows with suggestions that this star formation could be
triggered by the ionizing radiation (Sugitani et al., 2002; Billot et al.,
2010; Smith et al., 2010; Chauhan et al., 2011; Reiter & Smith, 2013; Cortes-
Rangel et al., 2019). However, the RDI picture of forming pillars raises the
question as to whether star formation is really ’triggered’, as stars could
have formed by the direct gravitational collapse of the pre-existing density
enhancements anyway, and the ionizing radiation need not have enhanced local
star formation in any way. In addition, numerical simulations have shown that
photoionization may actually have a negative global net effect on star
formation in GMCs, helping to explain the observed low star formation rates
(Vázquez-Semadeni et al., 2010; Dale et al., 2012; Colín et al., 2013;
Gavagnin et al., 2017; Geen et al., 2017; Kim et al., 2018). However,
photoionization feedback may also simultaneously trigger star formation
_locally_ , by increasing the number of stars formed (but not necessarily the
total mass of stars) (Dale et al., 2013). Moreover, Dale et al. (2015) argues
that current observational markers used to infer triggering may not be robust
enough to distinguish whether an observed star was triggered or has
spontaneously formed. Thus, there is no consensus on whether triggered star
formation is effective on GMC scales, and hence resolving and understanding
the local dynamics of the gas, in particular the turbulent flow is crucial for
determining the role of photoionization for star formation.

Turbulence regulates star formation in molecular clouds and can play a dual
role in the process: providing a form of support against self-gravitational
collapse due to random velocity fluctuations, and on the other hand forming
shocks resulting in overdensities that eventually undergo gravitational
collapse (Mac Low & Klessen, 2004). However, the observed supersonic
turbulence (Elmegreen & Scalo, 2004) must be driven by some external
mechanism, as it would otherwise decay within one crossing time (Mac Low et
al., 1998; Mac Low & Klessen, 2004; Stone et al., 1998). Various agents for
this driving have been proposed such as gravity, accretion, galactic rotation,
shearing flows, stellar feedback, etc. (see Federrath et al., 2017; Federrath,
2018, for a review of potential drivers), with the difference lying in the
type of turbulent modes that they drive: compressive (curl-free) or solenoidal
(divergence-free) modes. This difference is quantified through the driving
parameter $b$, which typically varies between 1/3 and 1, where these two
extreme cases refer to purely solenoidal and purely compressive driving,
respectively (Federrath et al., 2010a). The value of $b$ is important, as the
flow dynamics, density structure and the subsequent star formation rate depend
on it (Federrath et al., 2008; Federrath et al., 2010a; Price et al., 2011;
Konstandin et al., 2012; Padoan et al., 2014; Federrath & Banerjee, 2015;
Nolan et al., 2015); with compressive driving resulting in broader density
probability distribution functions (PDFs) and star formation rates
approximately an order of magnitude larger than for solenoidal driving
(Federrath & Klessen, 2012; Federrath et al., 2016; Federrath, 2018). The
values of $b$ have been studied systematically for different driving sources
of turbulence in numerical simulations (Pan et al., 2016; Körtgen et al.,
2017; Jin et al., 2017), and observations also find a significant variation in
$b$ across different clouds in the Milky Way (Padoan et al., 1997; Brunt,
2010; Ginsburg et al., 2013; Kainulainen, J. et al., 2013; Federrath et al.,
2016; Kainulainen & Federrath, 2017).

Expanding HII regions have been proposed to be one of the primary sources of
driving and maintaining supersonic turbulence in GMCs (Matzner, 2002; Krumholz
et al., 2006; Goldbaum et al., 2011). For instance, Gritschneder et al. (2009)
simulate the ionizing irradiation of a local patch of a cloud ($\sim
4\,\mathrm{pc}$) and compare it to control runs without irradiation where the
the initial turbulence is allowed to decay. They measure compressive,
solenoidal and total power spectra and find that turbulence is driven
significantly in the cold neutral gas, particularly in compressive modes, with
more efficient driving on smaller scales, leading to a flatter power spectrum.
However, there has been no systematic study of the driving parameter $b$ based
on density and velocity fluctuations for turbulence driven by expanding HII
regions. Studying this would allow for more direct inferences on whether the
local gas dynamics inside and around pillars shaped by the ionizing radiation
support a picture of triggered star formation or not.

In this study we calculate the driving parameter $b$ of ionizing radiation-
driven turbulence on the neutral gas in the vicinity of expanding HII regions.
We perform numerical simulations analogous to Gritschneder et al. (2009),
wherein we model the incoming radiation front as plane-parallel, and then
calculate $b$ from the previously established relation (Padoan et al., 1997;
Federrath et al., 2008; Federrath et al., 2010a; Price et al., 2011; Padoan &
Nordlund, 2011; Konstandin et al., 2012; Molina et al., 2012; Hopkins, 2013;
Federrath & Banerjee, 2015; Nolan et al., 2015; Squire & Hopkins, 2017; Mandal
et al., 2019),

$\sigma_{s}^{2}=\ln\left(1+b^{2}\mathcal{M}^{2}\right),$ (1)

where $\sigma_{s}$ is the standard deviation of the logarithmic density
contrast $s=\ln(\rho/\rho_{0})$ and the turbulent, sonic rms Mach number
($\mathcal{M}$). We further calculate the value of $b$ for sub-parsec-scale
regions at the tip of the formed pillar-like structures, where star formation
is expected to occur.

In Section 2 we define the simulation setup, initial condition, and the
numerical methods we use in the simulations. In Section 3 we present our
results and Section 4 provides a summary and conclusions.

##  2 Numerical methods

###  2.1 Physics and numerical methods

To model the evolution of the cloud and ionizing radiation, we follow the
equations of compressible three-dimensional hydrodynamics in combination with
an incoming plane-parallel photoionizing radiation flux in the $x$-direction
of the Cartesian computational domain. For the hydrodynamics, we utilize the
open source code _Pluto_ (Mignone et al., 2007, 2012) in version 4.1. For the
photoionization feedback, we make use of the _Sedna_ module, a ray-tracing
radiation transport solver, introduced in Kuiper & Hosokawa (2018). This
solver combination was also used for a recent code benchmark (Bisbas et al.,
2015). To generate the initial turbulent ISM setup, we implement the
turbulence generator described in Federrath et al. (2008) and Federrath et al.
(2010a).

###  2.2 Initial Conditions

Our simulation setup is largely similar to that of the fiducial simulation in
Gritschneder et al. (2009). The initial setup is a uniform Cartesian grid of
length $L=4\,\mathrm{pc}$ containing $1000\,\mathrm{M}_{\odot}$ of neutral gas
at a temperature of $T_{\mathrm{neutral}}=10\,\mathrm{K}$ corresponding to a
sound speed of $c_{\mathrm{s}}=0.28\,\mathrm{km}\,\mathrm{s}^{-1}$. To
initialise a neutral turbulent state, we introduce supersonic velocity
fluctuations with an rms Mach number of 10, i.e.,
$\mathcal{M}=\sigma_{v}/c_{\mathrm{s}}=10$, the ratio of turbulent velocity
dispersion and sound speed, with a velocity power spectrum following
$E(k)\propto k^{-2}$ (Federrath et al., 2010a; Federrath, 2013), consistent
with supersonic molecular cloud turbulence (Larson, 1981; Solomon et al.,
1987; Ossenkopf & Mac Low, 2002; Brunt & Heyer, 2002; Roman-Duval et al.,
2011). These velocity fluctuations are introduced in the wavenumber range
$2\leq k/(2\pi/L)\leq 20$, with a natural mixture of solenoidal and
compressive modes ($b\sim 0.4$; see Fig. 8 in Federrath et al., 2010a). This
is enforced by applying a projection in Fourier space, which decomposes the
driving field into its solenoidal and compressive components (Federrath et
al., 2008; Federrath et al., 2010a). We then allow the turbulence to develop
self-consistently by evolving the gas hydrodynamically with an isothermal
equation of state for one turbulent crossing time given by
$t_{\mathrm{crossing}}=L/\sigma_{v}$, where $\sigma_{v}$ is the velocity
dispersion. The computational volume boundaries are periodic in all spatial
directions during this phase. The turbulent state obtained after
$t=t_{\mathrm{crossing}}$ has a Mach number of $\sim 3.5$ due to the decay of
the initial turbulence. We show a column density projection of this stage in
Figure 1 (top-left panel), which serves as the initial condition for the
subsequent evolution of the cloud when it is hit by the ionization front
entering from the left-hand side of the $x$-axis. We define this stage as time
$t=0$ from here on.

###  2.3 Simulation Setup

After this turbulent state is reached ($t=0$), we introduce the ionizing
radiation of a massive star onto the turbulent gas. As we are interested in a
local region of a molecular cloud at the edge of the Strömgren sphere
(Strömgren, 1939), we approximate the ionization front as plane-parallel,
impinging from the negative $x$-direction in the simulation domain. The photon
flux per unit time is set to $\mathrm{F_{Ly}}=5\times
10^{9}\;\mathrm{photons}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}$. The ionization
cross section is set to $6.3\times 10^{-18}\,\mathrm{cm}^{2}$ and the
recombination rate into any state but the hydrogen ground state is set to
$2.6\times 10^{-13}\,\mathrm{cm}^{3}\,\mathrm{s}^{-1}$, making use of the so-
called on-the-spot approximation (Baker & Menzel, 1938; Spitzer, 1978;
Osterbrock, 1989). The ray tracer sets an ionization degree $\eta$ at each
cell from this flux, which is used to calculate the local temperature in the
cell by linear interpolation,

$T=T_{\mathrm{ion}}\times\eta+T_{\mathrm{neutral}}\times(1-\eta),$ (2)

where $T_{\mathrm{neutral}}=10\,\mathrm{K}$ and
$T_{\mathrm{ion}}=10^{4}\,\mathrm{K}$ are the neutral and ionized gas
temperatures, respectively (Shu, 1991). Both gas components can be
approximated as being in thermal equilibrium, since the heating and cooling
timescales are much shorter than the dynamical timescale. The sound speeds
$c_{\mathrm{s}}=\sqrt{k_{\mathrm{B}}T/(\mu m_{\mathrm{H}})}$ associated with
these phases are $0.28\,\mathrm{km}\,\mathrm{s}^{-1}$ and
$12.84\,\mathrm{km}\,\mathrm{s}^{-1}$, respectively, for atomic hydrogen gas
with $\mu=1.0$.

The fluid boundary conditions after switching on the radiation field are
periodic in all directions except the $x$-direction, where semi-permeable
walls are imposed, i.e., we allow the gas to leave the computational domain
through these boundaries, but do not allow new gas to enter. However, the
expansion of the HII region expels the neutral gas in the cloud through the
semi-permeable boundary at $x=4\,\mathrm{pc}$, compromising the study of
turbulence driving as the gas leaves the computational domain too early. To
delay this expulsion from the computational domain we account for the bulk
motion of the gas in the positive $x$-direction, by subtracting the
instantaneous centre of mass velocity from each cell’s local velocity at each
integration step. This allows us to follow the evolution of the cloud and
pillars in the rest frame of the system and for a sufficiently long time that
includes the formation, evolution and destruction (of some of) the pillars.

The adiabatic index ($\gamma$) of the ideal gas is set to $1.0001$, which
effectively establishes a locally isothermal state with the local temperature
set by the ionization degree of the gas. For simplicity, we assume the gas is
purely atomic with the mass per atom $m_{\mathrm{H}}\,=1.6\times
10^{-24}\,\mathrm{g}$. Note that we do not include magnetic fields, self-
gravity or a chemical network in our study. The grid resolution was set to
$200^{3}$ grid cells, and a single run took approximately 300 hours on 140
compute cores. However, we also provide a resolution study to compare our main
results for a grid resolution of $100^{3}$, $200^{3}$, and $400^{3}$ grid
cells, in Appendix A.

We note that the ray tracer solves the radiative transfer equation using the
on-the-spot (OTS) approximation. This neglects the effect of the diffuse
extreme ultraviolet (EUV) radiation emitted through recombinations in the
surrounding ionized gas. This is believed to be responsible for shadows seen
behind the pillars formed in other studies as well as our simulations and is
an artefact of the uni-directional ray tracing performed in such a simulation
setup. This leads to parcels of low-density gas lying in the shadows of these
pillars to remain neutral ($T\sim 10\,\mathrm{K}$), that would in reality be
at much higher temperatures due to the EUV recombination field from the
surrounding ionized gas. Numerical studies that include the diffuse radiation
field find that shadow regions with low density are at least partly ionized
and as a result the pillar structures are more compressed and less coherent
(Ercolano & Gritschneder, 2011; Haworth & Harries, 2012). We account for this
effect by excluding computational cells for which the temperature is
inaccurate given their density, from the analysis, as explained in the results
section below.

##  3 Results

###  3.1 Gas structure and evolution

The initial turbulent state is shown in Figure 1 (top-left panel). The
ionization front is impinging onto this turbulent gas from the lower
$x$-boundary of the computational domain (i.e., the $x=0$ face). It instantly
ionizes parts of gas in the cloud, with the radiation penetrating further into
low-density channels, forming a shock front at the transition from ionized to
neutral gas as can be seen at $t=200\,\mathrm{kyr}$ in the 2nd panel of Figure
1. On further evolution, the radiation continues to penetrate the low-density
channels but fails to ionize columns of gas in the $x$-direction that contain
overdensities seeded by the initial turbulent field. The ionized channels
expand and through their thermal pressure push the neutral gas in their
surroundings towards the columns of neutral gas in the shadows of
overdensities. This process compresses the gas tangentially to the direction
of the radiation and as a result, pillar-like structures form in the cold
neutral gas (top-right panel in Figure 1). Further evolution unveils
discernible pillar-like structures surrounded by a pool of hot ionized gas
reminiscent to those seen in observations (three selected pillar regions are
marked in the 2nd row of Figure 1). In addition to this thermal pressure-
induced compression, the surrounding ionized gas can push off the neutral
material, introducing a back-reaction known as the "rocket effect" that could
exert significant forces and excite turbulent modes in the neutral gas
(Krumholz et al., 2019). Note again that we are following the evolution in the
rest frame of the entire system, which is why some of the pillars appear to
move against the radiation direction.

Figure 1 shows that the tips of the pillars have the highest density
enhancements. It is at these pillar tips that star formation is observed to
occur (e.g., Smith et al., 2010; Reiter & Smith, 2013; Klaassen et al., 2014;
Klaassen et al., 2019). Thus, we define three different pillar regions for
more detailed analysis below: Pillar A, B and C of sizes $0.4\,\mathrm{pc}$ in
each direction. The regions are picked by eye from their line-integrated
column densities in each direction while ensuring that the high-density tip of
the pillar is included, and then adjusted such that the centre of mass of the
region lies at the centre of the defined cubical box. These defined regions
are shown for the line-integrated column density in the $z$-direction in
Figure 1 (2nd row and marked with boxes) for three different time snapshots
during which we follow their evolution.

At later stages of the evolution ($t>1000\,\mathrm{kyr}$) the ionizing
radiation succeeds in photo-evaporating or expelling a large fraction of
neutral gas in the cloud. However, following the system in the instantaneous
centre of mass frame of reference allows us to see the late time evolution of
some pillars and isolated globules that manage to survive. Their survival can
be attributed to the strong turbulent ram pressure in their interiors with
comparable strengths to the thermal pressure of the ionized gas. Observations
have revealed such structures in the vicinity of HII regions and they are
usually categorised as globules, evaporating gaseous globules (EGGs),
condensations and proplyds based on their morphology and sizes, and are
proposed to eventually form from radiation sculpted pillar-like structures
(see for instance Schneider et al., 2016).

Figure 1: Time evolution of the column density of the gas integrated along the
$z$-direction. The UV ionizing radiation enters the computational volume from
the $x=0$ face. The radiation ionizes parts of the cloud that sculpts pillar-
like columns of neutral gas with the densest parts at its tips. The boxes in
the 2nd row of panels indicate three sub-parsec scale ($\sim 0.4\mathrm{pc}$)
regions containing the tips of pillars where we individually analyse the
turbulence driving mode.

###  3.2 Driving Parameter $b$

The driving parameter $b$ is a quantity that is proportional to the ratio of
density to velocity fluctuations, $b\propto\sigma_{\rho}/\sigma_{v}$ in a
supersonically turbulent cloud (Federrath et al., 2008; Federrath et al.,
2010a). A velocity field that contains primarily compressible modes would
produce stronger compressions and rarefactions and thus result in a higher
spread in the density PDF than a primarily solenoidal velocity field. From the
well-studied relation given by Equation (1) between the log-density dispersion
$\sigma_{s}$ and the rms turbulent Mach number $\mathcal{M}$, we can rearrange
for the driving parameter $b$,

$b=\left\\{\left[\exp\left(\sigma_{s}^{2}\right)-1\right]/\mathcal{M}^{2}\right\\}^{1/2}.$
(3)

Calculating $b$ for an isothermal gas is fairly straightforward as the sound
speed of isothermal gas is constant and identical everywhere, and
$\mathcal{M}$ would just be the turbulent rms velocity dispersion
($\sigma_{v}$) divided by that global sound speed. This condition is satisfied
for time $t=0\,\mathrm{kyr}$ in our simulation where all the gas is at $T\sim
10\,\mathrm{K}$, and we obtain a value of $b\sim 0.43$; which is the value
expected for isothermal gas initialised with turbulent velocity fluctuations
having a natural mixture of compressive and solenoidal modes (Federrath et
al., 2010a).

However, for $t>0$ the situation is more complicated as there are a range of
local sound speeds due to the ionizing radiation, and thus a range of local
Mach numbers ($M$). Thus, we have to distinguish the rms Mach number
$\mathcal{M}$ from the local Mach number $M$. Strictly speaking, Equation (3)
is only valid for an isothermal gas, as it has been derived and tested under
isothermal conditions, and we can thus only calculate $b$ for the cold neutral
gas in our simulations, i.e., by selecting regions of gas belonging to the
cold neutral phase, and calculating the value of $b$ for them.

####  3.2.1 Determining $\sigma_{s}$ and $\mathcal{M}$ for the dense, cold gas

In this study we are concerned with the turbulence in the star-forming neutral
gas that is driven by the surrounding ionized gas when irradiated with
ionizing radiation. Hence we first pick out regions in our computational
volume that have ionization fractions corresponding to neutral gas, i.e.,
$\eta<10^{-7}$, which effectively corresponds to gas with temperature equal to
the neutral gas temperature (i.e., $T=10\,\mathrm{K}$).

In Figure 2 we show a scatter plot for $t=800\,\mathrm{kyr}$ of the local
values of the scaled logarithmic density ($s$) and the Mach number ($M$), with
the colourbar representing the mass-weighted probability density. We notice
that over the wide range of $s$ and $Mach$ covered in the simulation, there is
an anti-correlation between $s$ and $M$, which is a result of the fact the gas
is not globally isothermal (Federrath & Banerjee, 2015). Moreover, the
geometry of the pillar-like structures, with a high positive gradient in the
density away from the ionized gas towards the confined dense columns, results
in the lower-density edges of the pillars having higher velocities as compared
to the denser interiors, leading to the anti-correlation. In contrast,
Equation (3) only holds for the dense, cold, near-isothermal gas, where $s$
and $M$ are un-correlated (Passot & Vázquez-Semadeni, 1998; Kritsuk et al.,
2007; Federrath et al., 2010a).

In addition to this we notice that some of these low-density regions have
extremely high local Mach numbers, with some of them exceeding $M\gtrsim 100$.
Careful analysis of the data reveals that they lie at the shadows of the dense
pillar tips, at the transition between the ionized and neutral phases of the
gas, and the high velocities are the result of the large pressure gradient.
However, as mentioned earlier, the cool temperatures of these regions ($T\sim
10\,\mathrm{K}$) are numerical artefacts due to the uni-directional ray-tracer
used and the lack of a diffuse EUV recombination field and would in reality
have higher ionization fractions (and hence higher $c_{s}$), which would
effectively reduce their $M$ to lower and more realistic values. To avoid the
bias introduced by these artefacts, we 1) apply a density threshold of
$n>n_{\mathrm{threshold}}=10^{2}\,\mathrm{cm}^{-3}$ and 2) use mass-weighted
PDFs, such that the low-density regions with unrealistic values of $M$ do not
contribute significantly to the density and velocity moments that enter Eq.
(3). This method allows us to measure the turbulence driving parameter $b$ for
the dense, cold phase of neutral gas, where star formation would occur.

The flanking histograms in Figure 2 show the mass-weighted PDFs for $s$
(integrated over $M$) and $M$ (integrated over $s$). The density threshold
$n>n_{\mathrm{threshold}}=10^{2}\,\mathrm{cm}^{-3}$ is indicated as a vertical
dotted line, where $n_{\mathrm{threshold}}$ is the number density of the
transition from atomic to molecular gas via surface reactions on dust grains.
This value has been studied to lie in the range $100-1000\,\mathrm{cm}^{-3}$
(Glover & Mac Low, 2007; Glover et al., 2010). Here we simply use
$n_{\mathrm{threshold}}=10^{2}\,\mathrm{cm}^{-3}$, however, we also repeat our
analyses for $n_{\mathrm{threshold}}=1000\,\mathrm{cm}^{-3}$ in Appendix B and
show that our results do not change significantly with the chosen density
threshold. The shaded flanking PDFs overlaid on the overall neutral gas PDFs
in Figure 2 show the distributions with the density threshold applied.

We now derive values for $\sigma_{s}$ and $\mathcal{M}$ from the distributions
of $s$ and $M$, respectively. For the distribution of $s$ we fit the mass-
weighted version of a lognormal given by Li et al. (2003),

$P_{\mathrm{LN}}(s)\,ds=\frac{\mathrm{C}\exp(s)}{\sqrt{2\pi\sigma_{s,\mathrm{MW}}^{2}}}\exp{\left[-\frac{s-s_{0}+\sigma_{s,\mathrm{MW}}^{2}/2}{2\sigma_{s,\mathrm{MW}}^{2}}\right]}\,ds,$
(4)

where $s_{0}$ is mean value of $s$, $\sigma_{s,\mathrm{MW}}$ is the mass-
weighted standard deviation in $s$, and $\mathrm{C}$ is a normalisation
constant. We free all three parameters in this relation and compute their best
fit values using a least-squares approach. We then set $\sigma_{s}$ to be
equal to the best fit value of $\sigma_{s,\mathrm{MW}}$. To derive
$\mathcal{M}$, we explicitly calculate the mass-weighted standard deviation of
the local Mach number distribution given by

$\mathcal{M}=\sqrt{\langle M_{\mathrm{MW}}^{2}\rangle-\langle
M_{\mathrm{MW}}\rangle^{2}},$ (5)

where $\langle M_{\mathrm{MW}}^{2}\rangle$ and $\langle
M_{\mathrm{MW}}\rangle$ are the mean squared and arithmetic mean of the mass-
weighted $M$, respectively. Figure 2 demonstrates this method of analysis for
the time snapshot $t=800\,\mathrm{kyr}$, with the derived values of
$\sigma_{s}$ from the lognormal fit and $\mathcal{M}$ for the dense, cold gas
denoted on the flanking histograms. This method is repeated for each time
snapshot to obtain the time evolution of $b$.

Figure 2: Scatter plot of local Mach numbers ($M$) and scaled logarithmic
densities, $s=\ln(\rho/\rho_{0})$, for the neutral gas in the entire
simulation domain at time $t=800\,\mathrm{kyr}$, colour-coded by the mass-
weighted probability density. Flanking histograms show the respective
individual mass-weighted PDFs (grey) and the same for
$n>10^{2}\,\mathrm{cm}^{-3}$ (shaded blue). The derived value of $\sigma_{s}$
from a lognormal fit (Eq. 4, shown in orange) and $\mathcal{M}$ from the mass-
weighted standard deviation of $M$ (computed from Eq. 5) are denoted in the
respective panels. The values of $\sigma_{s}$ and $\mathcal{M}$ obtained with
this method are then used in Equation 3 to compute the driving parameter $b$.

####  3.2.2 Time evolution of the global driving parameter $b$

Figure 3 shows the time evolution of $b$ with the method outlined in the
previous subsection. We divide this time evolution into three broad temporal
phases:

  * •

Transient Phase ($0\leq t<300\,\mathrm{kyr}$): the gas ionized by the
radiation field is in the process of enhancing the initial overdensities
seeded by the turbulence and sculpting pillar-like regions. The turbulence
driven by the thermal pressure-induced compression of neutral gas, as evident
from a sharply increasing value of $\mathcal{M}$, pushes neutral gas from
lower density columns into denser ones, resulting in an enhancement of the
initial overdensities and increasing $\sigma_{s}$. We observe a time lag in
the increase of $\sigma_{s}$ and $\mathcal{M}$, as one is the cause (higher
$\mathcal{M}$) and the other the effect (higher $\sigma_{s}$), and this lag
biases the value of $b$ to be lower at very early times (up to $\sim
200\,\mathrm{kyr}$). When the velocity fluctuations start to succeed in
enhancing the initial overdensities by dissipating their energy in shocks,
$\mathcal{M}$ starts to decrease until it reaches a steady value of
$\mathcal{M}\sim 5.5$.

  * •

Driving Phase ($300\leq t<1100\,\mathrm{kyr}$): pillar-like structures have
already been created as a result of the ionizing radiation-driven compression
and the dynamics of the neutral gas have been established, albeit subject to
local structural changes due to compression or photo-evaporation at the
individual pillar regions (as seen from the temporal fluctuations in
$\sigma_{s}$). We thus call this phase the ’Driving Phase‘, as the turbulence
has settled down and thus a driving parameter $b$ can be reasonably defined
here. As we can see in Figure 3, a fairly stationary value of $b$ is obtained
in this phase. We calculate the time-averaged value of $b$ in this phase as
$\langle b\rangle_{t}=0.76\pm 0.08$.

  * •

Disruption Phase ($1100\leq t<1500\,\mathrm{kyr}$): at these late times of the
evolution, many of the neutral pillar-like structures get either expelled
through the outflow boundaries of the computational box and/or get
successfully ionized by the expanding HII region. The amount of neutral
molecular gas available in the cloud is insufficient to derive meaningful
statistics, as is evident from the relatively large error bars in this final
phase of evolution.

We thus argue that the value of $b$ obtained in what we call the ’Driving
Phase‘ most sensibly represents what could be characterised as the turbulence
driving mode of ionizing radiation. The time-averaged $b\sim 0.76$ suggests
that the turbulence driven in the neutral molecular gas by the ionizing
radiation on cloud scales ($\sim 4\;\mathrm{pc}$) is predominantly compressive
($b>0.4$) in nature.

Figure 3: Evolution of the driving parameter $b$ and the quantities involved
in its calculation with time, where $t=0$ is the time when the ionization
front hits the gas. Shaded regions indicate the phase of evolution as
categorised in Section 3.2.1. From top to bottom: logarithmic scaled density
dispersion $\sigma_{s}$, rms Mach number $\mathcal{M}$, and driving parameter
$b$. The dotted line in red indicates the time-averaged value of $b$ in the
Driving Phase $\langle b\rangle_{t}=0.76\pm 0.08$, and the values of $b$
corresponding to purely compressive (green) and purely solenoidal (violet)
modes are also provided for reference, as dotted lines. The values of the same
quantities obtained for the defined pillar regions (c.f. Fig. 1), A
(diamonds), B (circles), and C (squares), are shown for 5 different time
snapshots, where these could be reasonably defined and followed in their
individual evolution. Figure 4: Same as Figure 2, but for the sub-parsec
region Pillar C at $t=800\,\mathrm{kyr}$, demonstrating our analysis method to
derive values of $b$ for the three pillar regions. We do not use a density
cutoff here as the contributions from $n<100\;\mathrm{cm}^{-3}$ are
negligible. We use the Hopkins (2013) intermittency PDF model (Eq. 6) instead
of a standard lognormal to fit for $\sigma_{s}$ (shown in orange).

###  3.3 Pillar Regions

Studies show that star formation is observed at the tips of pillar-like
structures in a molecular cloud irradiated by ionizing radiation. Information
on the local dynamics of the region is thus important to understand the star
formation potential of the gas. Hence we now calculate the driving parameter
$b$ separately for the three sub-parsec-scale pillar regions denoted in Figure
1, namely Pillar A, B and C. The regions all have cubical sizes of
$(0.4\,\mathrm{pc})^{3}$, adjusted such that the centre of mass of the region
lies at the centre of the defined cubical box. We then select again only the
neutral gas in these regions ($\eta<10^{-7}$). However, we do not set any
density threshold for pillar regions as the contribution of gas with
$n<n_{\mathrm{threshold}}$ is negligible. We derive the Mach number
($\mathcal{M}$) as we did for the global gas, i.e., as the standard deviation
of the mass-weighted distribution of the local Mach number ($M$), and given by
Equation 5. However, the distribution of $s$ is more skewed, and a symmetrical
lognormal does not fit the data well on the local pillar scales, most likely
because of intermittency in the turbulence (Kritsuk et al., 2007; Federrath et
al., 2010a; Hopkins, 2013; Squire & Hopkins, 2017). To account for this
skewness, we instead fit the mass-weighted version of the Hopkins (2013)
intermittency PDF model,

$p_{\mathrm{HK}}(s)\,ds=\mathrm{C}e^{s}I_{1}(2\sqrt{\lambda\omega(s)})\exp{\left[-(\lambda+\omega(s))\right]}\sqrt{\frac{\lambda}{\theta^{2}\omega(s)}}\,ds,\\\
\lambda=\frac{\sigma_{s,\mathrm{MW}}^{2}(1+\theta)^{3}}{2\theta^{2}},\quad\omega(s)=\lambda/(1+\theta)-s/\theta\;(\omega\geq
0),$ (6)

where $I_{1}(x)$ is the first-order modified Bessel Function of the first
kind, $\sigma_{s,\mathrm{MW}}$ is the mass-weighted standard deviation in $s$,
and $\theta$ is the intermittency parameter. Note that in the zero-
intermittency limit ($\theta\to 0$) Eq. (6) simplifies to the lognormal PDF.
We fit Equation 6 to $P_{\mathrm{MW}}(s)$ and determine the best-fit
$\sigma_{s,\mathrm{MW}}$. We show the scatter of $s$ and $M$, their
corresponding mass-weighted histograms, and the values of $\sigma_{s}$ and
$\mathcal{M}$ in Figure 4 for time snapshot $t=800\,\mathrm{kyr}$. These are
then used in Equation 3 to obtain the value of $b$. This is repeated for five
different time snapshots in the Driving Phase and the obtained values were
added in Figure 3.

Figure 3 shows that the values for $\sigma_{s}$ and $\mathcal{M}$ for the
pillar regions are lower than that for the gas on the global cloud scales.
This is expected from the observed cloud scaling relations, such as the
velocity dispersion–size relation. The pillar regions are considerably smaller
($\sim 0.4\,\mathrm{pc}$) and thus have only mildly supersonic velocity
dispersions ($\mathcal{M}\sim 2$–$4$). The obtained velocity dispersions
($\mathcal{M}\times c_{\mathrm{s}}$) are in the range
$0.4$–$1\,\mathrm{km}\,\mathrm{s}^{-1}$, in agreement with observations of
pillars on these scales (Klaassen et al., 2019).

The $b$ values of the pillar regions are also somewhat smaller with $b\sim
0.4$–$0.6$ compared to the cloud-scale average. Interestingly, we find a
noticeable time variation for each pillar region, individually. This is more
clearly seen in Figure 5, where we find that the value of $b$ increases for
each pillar, from a value of $b\sim 0.4$ to $0.55$ as time progresses,
indicating that the turbulence in these regions develops more compressive
modes as they evolve. This might suggest that conditions for star formation
are progressively boosted in these regions, as the HII region moves through
the cloud. That being said, commenting further on the potential star formation
in these simulations is outside the scope of this work as it would require
follow-up simulations that include self-gravity and a model for star formation
such as the sink-particle technique (e.g., Federrath et al., 2010b).

Figure 5: Zoom-in of the time evolution of the driving parameter $b$ during
the Driving Phase for the three pillar regions. Dotted lines indicate purely
compressive (green) and purely solenoidal (brown) values of $b$ for reference.
Solid lines indicate the time evolution of $b$ for the gas on $\sim
4\,\mathrm{pc}$ global cloud scales (blue) and the value of $b$ inferred from
the compressive ratio $b_{\chi}$ (red) discussed in Section 3.4.

###  3.4 The compressive ratio

So far we have primarily used the density dispersion and rms Mach number to
study the turbulence driving parameter $b$ of clouds hit by an ionization
front. However, we can also look at the compressible-to-solenoidal mode
mixture in the velocity field to obtain related information about $b$, via the
so-called ’compressive ratio‘ (Kritsuk et al., 2007; Federrath et al., 2010a,
2011; Pan et al., 2016; Körtgen et al., 2017). The compressive ratio is given
by

$\chi=\langle v_{c}^{2}\rangle/\langle v_{s}^{2}\rangle,$ (7)

where $v_{c}$ and $v_{s}$ are the compressive and solenoidal components of the
velocity field, respectively. In order to compute these two components, the
velocity field is first corrected for bulk motion, such that only the
turbulent fluctuations remain (analogous to $\chi_{\mathrm{turb}}$ defined in
Pan et al., 2016), and then density-weighted to trace the velocity field of
the cold, dense gas. Both $v_{c}$ and $v_{s}$ are then derived through a
Helmholtz decomposition of the turbulent velocity field in Fourier space.
Following the suggestion of Pan et al. (2016), an effective driving parameter
can be derived from this value of $\chi$ given by
$b_{\chi}=\sqrt{\chi/(\chi+1)}$.

The evolution of $b_{\chi}$ during the Driving Phase is shown in Figure 5. As
we can see the value we obtain remains more or less constant ($\sim 0.75$) and
indicates strong compressive velocity fields. This value is also very close to
the time-averaged value $\langle b\rangle_{t}$ obtained earlier from the
relation between the density and velocity fluctuations (Equation 3). The fact
that $b$ obtained from the $\sigma_{s}$–$\mathcal{M}$ relation (Eq. 1) agrees
with that obtained from the modes in the velocity field ($b_{\chi}$) is
encouraging and supports our overall conclusions of a relatively compressive
turbulence driving parameter ($b>0.4$) for clouds compressed in HII regions.

###  3.5 Virial Parameter

The virial parameter is a dimensionless quantity that characterises the ratio
of turbulent kinetic energy to gravitational energy of a cloud of gas
(Bertoldi & McKee, 1992), with its general form given by (Federrath & Klessen,
2012),

$\alpha_{\mathrm{vir}}=2E_{\mathrm{kin}}/|E_{\mathrm{grav}}|,$ (8)

where $E_{\mathrm{kin}}$ and $E_{\mathrm{grav}}$ are the kinetic and potential
energies of the cloud, respectively. A value of $\alpha_{\mathrm{vir}}<1$
suggests that the cloud could be gravitationally unstable and thus potentially
form stars, with the star formation rate per free fall time
$\mathrm{SFR}_{\mathrm{ff}}$ increasing with decreasing
$\alpha_{\mathrm{vir}}$ (Krumholz & McKee, 2005; Hennebelle & Chabrier, 2011;
Padoan et al., 2012; Federrath & Klessen, 2012).

We therefore study the time evolution of $\alpha_{\mathrm{vir}}$ for our
defined pillar regions to test whether the driving of compressive turbulence
is accompanied by an increase in the star-formation capability of the gas. We
calculate $\alpha_{\mathrm{vir}}$ by computing the self-gravitational
potential $\Phi$ for each pillar region from their respective density
distributions, with boundary conditions taking into account the density
distribution of gas outside the boundaries of the pillar regions. This is
important, because the binding energy of pillars is not independent of the
environment (Federrath & Klessen, 2012), as these regions are not isolated,
but instead are typically deeply embedded inside a larger molecular cloud
complex. We calculate $\alpha_{\mathrm{vir}}$ as

$\alpha_{\mathrm{vir}}=\frac{\sum_{i\in\mathrm{P}}m_{i}|(\mathbf{v}_{\mathrm{turb}})_{i}|^{2}}{\sum_{i\in\mathrm{P}}m_{i}|\Phi_{i}|},$
(9)

where $\mathrm{P}$ is the set of computational cells lying in the pillar
region, $m_{i}$ the mass, and $(\mathbf{v}_{\mathrm{turb}})_{i}$ the turbulent
velocity in cell $i$. The time evolution of this calculated
$\alpha_{\mathrm{vir}}$ for the pillars is shown in Figure 6.

Figure 6: Time evolution of the virial parameter $\alpha_{\mathrm{vir}}$
computed with Eq. (9) for each of the three pillar regions.

As we can see the value of $\alpha_{\mathrm{vir}}$ goes down with time for all
the pillar regions in our simulations. This is because the turbulent kinetic
energy in the pillars is dissipated in shocks produced by the compressive
turbulence (decreasing $|(\mathbf{v}_{\mathrm{turb}})_{i}|$ locally). This
leads to the formation of local overdensities in the pillars (increasing
$|\Phi_{i}|$ locally), which effectively results in lower values of
$\alpha_{\mathrm{vir}}$. This suggests that the driving of compressive
turbulence in the pillars lowers the value of the virial parameter and as a
result, could potentially trigger star formation in them. We note that there
are alternatives to Equation (9) to calculate the virial parameter, especially
in observational studies where the self-gravitational potential is not
available. In Appendix C we test these different methods, and find that
although there is a significant difference in the absolute values of
$\alpha_{\mathrm{vir}}$ obtained with different methods, the trend of a
decreasing $\alpha_{\mathrm{vir}}$ with time is independent of the choice of
method.

##  4 Summary and conclusions

In this study we simulate the formation of pillar-like structures in turbulent
clouds at the edge of a Strömgren sphere, by an expanding HII region modelled
as an impinging plane-parallel ionizing front. We form structures that
resemble observed pillars in morphology and dynamics, and study their
evolution for up to $2\,$ Myr. We infer the effective mode of turbulence that
is driven by the ionization front in the dense, cold gas of the cloud, by
calculating the turbulence driving parameter ($b$) in Equation (1). We do the
same for three different sub-parsec scale regions at the tip of pillars
containing the densest gas, where star formation is often observed. Our main
results can be summarised as follows:

  1. 1.

The value of $b$ for the cold, dense gas in the cloud does not vary
significantly with time during the Driving Phase ($300$–$1100\,\mathrm{kyr}$),
with a time-averaged value of $\langle b\rangle_{t}=0.76\pm 0.08$. This
suggests that expanding HII regions drive predominately compressive modes of
turbulence.

  2. 2.

The values of $b$ calculated for the pillar regions are also primarily
compressive albeit with lower values of $b\sim 0.4$–$0.6$ compared to the gas
on global cloud scales. We also obtain values for the 3D velocity dispersion
($\sigma_{v}\sim 0.4$–$1\,\mathrm{km}\,\mathrm{s}^{-1}$) that agree with
observations in the pillar regions.

  3. 3.

The turbulent modes for each of the pillar regions transition from a natural
mixture of solenoidal and compressive modes ($b\sim 0.4$) to a compression-
dominated regime ($b\sim 0.55$) during their lifetime, as the ionized gas
continues to sculpt the pillars.

  4. 4.

We calculate the compressive ratio ($\chi$) for the cloud and infer a driving
parameter $b_{\chi}\sim 0.75$ from it, consistent with the time-averaged
global $b$ value obtained from the density dispersion–Mach number relation,
supporting our main conclusion that expanding HII regions drive primarily
compressive modes of turbulence.

  5. 5.

The virial parameter $\alpha_{\mathrm{vir}}$ decreases with time for the
pillar regions, which suggests that the driving of compressive turbulence in
these regions, as the HII region passes through the cloud, is accompanied by
an increase in the star-formation capability of the gas.

Our main finding of predominantly compressive turbulence driven in HII regions
may be interpreted as promoting star formation, and hence leading to
’triggered star formation’. However, the converse effect of the neutral gas
being photo-evaporated by the radiation limits the extent of this triggering,
and the net star formation occurring in the region is likely the result of the
competition between these two processes. Follow-up simulations with self-
gravity and sink particles to follow star formation would allow us to make
more quantitative predictions on the net effect of photo-ionization on star
formation in molecular clouds.

## Acknowledgements

We thank Richard Wunsch for a timely and very constructive referee report. SHM
and RK acknowledge financial support via the Emmy Noether Research Group on
Accretion Flows and Feedback in Realistic Models of Massive Star Formation
funded by the German Research Foundation (DFG) under grant no. KU 2849/3-1 and
KU 2849/3-2. C. F. acknowledges funding provided by the Australian Research
Council (Discovery Project DP170100603 and Future Fellowship FT180100495), and
the Australia-Germany Joint Research Cooperation Scheme (UA-DAAD). We
acknowledge support by the High Performance and Cloud Computing Group at the
Zentrum für Datenverarbeitung of the University of Tübingen, the state of
Baden-Württemberg through bwHPC and the German Research Foundation (DFG)
through grant no INST 37/935- 1 FUGG. We further acknowledge high-performance
computing resources provided by the Leibniz Rechenzentrum and the Gauss Centre
for Supercomputing (grants pr32lo, pr48pi and GCS Large-scale project 10391),
the Australian National Computational Infrastructure (grant ek9) in the
framework of the National Computational Merit Allocation Scheme and the ANU
Merit Allocation Scheme.

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##  Appendix A Resolution Study

The decay rate of the turbulent energy and the general properties of
turbulence in numerical simulations depend on the numerical resolution
adopted, with lower resolution showing, e.g., a larger (artificial) decay (Mac
Low et al., 1998; Federrath, 2013). In order to test numerical convergence of
our results, we compare $\sigma_{s}$, $\mathcal{M}$ and the resulting $b$
parameter (from Eq. 3) for three different numerical grid resolutions:
$100^{3}$ ($\Delta x=0.04\,\mathrm{pc}$), $200^{3}$ ($\Delta
x=0.02\,\mathrm{pc}$), and $400^{3}$ ($\Delta x=0.01\,\mathrm{pc}$). We note
that the $400^{3}$ simulation has only been followed for comparison purposes
up to $\sim 800\,\mathrm{kyr}$ due to the computational cost of that
simulation. Figure 7 shows the scatter plot and corresponding PDFs of $s$ and
$M$ for the three different resolutions at $t=700\,\mathrm{kyr}$, and Figure 8
compares the time evolution of $\sigma_{s}$, $\mathcal{M}$ and $b$ derived
from the PDFs, following the method explained in the main part of the
manuscript. We can see that $\mathcal{M}$ is almost independent of the choice
of resolution (as long as the grid resolution is at least $100^{3}$). However,
$\sigma_{s}$ and $b$ show a systematic increase between $100^{3}$ and
$200^{3}$, while for $\gtrsim 200^{3}$, the resulting $\sigma_{s}$ and $b$ are
close to the values obtained for $200^{3}$ grid cells. Similar trends for the
dependence of $\mathcal{M}$ and $\sigma_{s}$ are seen in Federrath et al.
(2010a), Kitsionas et al. (2009), and later resolution studies. We thus
conclude that a spatial resolutions of $\lesssim 0.02\,\mathrm{pc}$ is
sufficient to achieve reasonable convergence in this type of simulation.

Figure 7: Scatter plot and flanking histograms (similar to Figure 2) for
$t=700\,\mathrm{kyr}$ comparing three different numerical grid resolutions.
Top-left : $100^{3}$ ($\Delta x=0.04\,\mathrm{pc}$), top-right : $200^{3}$
($\Delta x=0.02\,\mathrm{pc}$) and bottom: $400^{3}$ ($\Delta
x=0.01\,\mathrm{pc}$). Figure 8: The values of $\sigma_{s}$, $\mathcal{M}$,
and $b$ compared for simulations with resolutions of $100^{3}$ ($\Delta
x=0.04\,\mathrm{pc}$), $200^{3}$ ($\Delta x=0.02\,\mathrm{pc}$) and $400^{3}$
($\Delta x=0.01\,\mathrm{pc}$). We find reasonable convergence for resolutions
of $\gtrsim 200^{3}$ grid cells ($\Delta x\lesssim 0.02\,\mathrm{pc}$).

##  Appendix B Choice of density threshold

In this study we use a threshold for the number density,
$n_{\mathrm{threshold}}=10^{2}\,\mathrm{cm}^{-3}$, for selecting dense, cold
gas, representing the minimum number density where the transition from atomic
to molecular gas via surface reactions on dust grains occurs. This value is
expected to be in the range $100-1000\;\mathrm{cm}^{-3}$ as shown in e.g.,
Glover & Mac Low (2007), and thus we adopt a fiducial value of
$n_{\mathrm{threshold}}=10^{2}\,\mathrm{cm}^{-3}$ for our main analyses.
However, here we show that our results do not significantly depend on the
exact choice of $n_{\mathrm{threshold}}$, by comparing results for
$n_{\mathrm{threshold}}=10^{2}\,\mathrm{cm}^{-3}$ to
$10^{3}\,\mathrm{cm}^{-3}$ in Figure 9. Apart from the initial transient phase
(where the amount of gas satisfying the condition $n>1000\,\mathrm{cm}^{-3}$
is very low and hence the error bars are very high), the values of $b$ for
both density thresholds agree to within $<10\%$ deviation. The time-averaged
values of $b$ in the Driving Phase are $0.76\pm 0.08$ and $0.81\pm 0.12$ for
$n_{\mathrm{threshold}}=10^{2}\,\mathrm{cm}^{-3}$ and
$10^{3}\,\mathrm{cm}^{-3}$, respectively. We can thus conclude that adopting
any value for $n_{\mathrm{threshold}}$ in the realistic range of the
transition number density ($100$–$1000\,\mathrm{cm}^{-3}$) gives robust
results.

Figure 9: Comparison of derived values of $\sigma_{s}$, $\mathcal{M}$, and $b$
with $n_{\mathrm{threshold}}=100\,\mathrm{cm}^{-3}$ and
$1000\,\mathrm{cm}^{-3}$. We find that our results are not particularly
sensitive to the choice of $n_{\mathrm{threshold}}$ as long as
$n_{\mathrm{threshold}}$ is in the reasonable range
$10^{2}$–$10^{3}\,\mathrm{cm}^{-3}$ to define dense, cold gas.

##  Appendix C Virial parameter method

In Section 3.5 we compute the virial parameter ($\alpha_{\mathrm{vir}}$) for
the pillars by calculating the ratio of their kinetic and potential energies,
which are individually estimated as the sum of the local contributions to the
energies by computational cells belonging to the region of interest. However,
there are alternative methods used to calculate $\alpha_{\mathrm{vir}}$, and
the values obtained may depend on the method. To test the dependence of
$\alpha_{\mathrm{vir}}$ on the method, we explore two alternative methods to
calculate the virial parameter, namely

  * •

using a fixed velocity dispersion. In this, case we do not take into account
the individual contributions of cells in the calculation of the kinetic
energies, but rather estimate it from the velocity dispersion ($\sigma_{v}$)
for the entire pillar region. $\alpha_{\mathrm{vir}}$ is then

$\alpha_{\mathrm{vir}}=\frac{\sigma_{v}^{2}\sum_{i\in\mathrm{P}}m_{i}}{\sum_{i\in\mathrm{P}}m_{i}|\Phi_{i}|},$
(10)

  * •

using an approximation that treats the pillar as if it were an isolated,
spherical region. For this method, we assume the pillars can be approximated
as spherical clouds of mass $M_{c}=\sum_{i\in\mathrm{P}}m_{i}$ and radius
$R_{c}$ occupying a volume $V=4/3\pi R_{c}^{3}=\sum_{i\in\mathrm{P}}V_{i}$,
where $V_{i}$ is the volume occupied by computational cell $i$.
$\alpha_{\mathrm{vir}}$ is then estimated as

$\alpha_{\mathrm{vir}}=\frac{5\sigma_{v}^{2}L}{6GM_{c}},$ (11)

where $L=2R_{c}$, $G$ is the gravitational constant, and $\sigma_{v}$ is the
velocity dispersion in the pillar. This is the method widely used in
observational studies to determine $\alpha_{\mathrm{vir}}$, because the 3D
gravitational potential is not available in observations.

In Figure 10 we compare the values of $\alpha_{\mathrm{vir}}$ obtained with
the above methods and the method used in Section 3.5. We find that
$\alpha_{\mathrm{vir}}$ decreases with time for the pillars in all three
methods, which suggests that our basic conclusions regarding the time
evolution of the pillars is independent of the choice of method. However, we
note that the values of $\alpha_{\mathrm{vir}}$ vary significantly with the
method adopted, especially in the isolated, spherical-cloud method. This is
because the definition of $\alpha_{\mathrm{vir}}$ in this method is based on
global parameters, assuming a spherical, homogeneous cloud. This is far from
realistic, as the pillars are highly non-homogeneous and non-spherical. The
gas in the pillars is turbulent, and hence can be concentrated locally in
fractal structures that cause a decrease in $\alpha_{\mathrm{vir}}$ without
necessarily increasing the total mass in the region. In addition, the
spherical cloud is assumed to be isolated, and hence the dynamical effects on
the gravitational potential due to the gas outside the boundary of the pillar
is not considered (Federrath & Klessen, 2012). The large difference in the
value of $\alpha_{\mathrm{vir}}$ obtained with this approximation suggests
that it should be used and interpreted with caution, and may not necessarily
represent the true dynamical state of the gas in the cloud.

Figure 10: Evolution of the virial parameter $\alpha_{\mathrm{vir}}$ with time
obtained for the pillar regions using three different methods: 1) the method
defined in Section 3.5, which uses all local information to compute
$\alpha_{\mathrm{vir}}$ (top panel), 2) using a fixed velocity dispersion
(middle panel), given by Equation (10), and 3) using the isolated, spherical-
cloud approximation (bottom panel), given by Equation (11). We find that our
result of a decreasing trend for $\alpha_{\mathrm{vir}}$ with time is robust,
however, the value of $\alpha_{\mathrm{vir}}$ changes significantly with the
choice of method.


# Multiresolution Textual Inversion

Giannis Daras
Department of Computer Science
UT Austin
<EMAIL_ADDRESS>
&Alexandros G. Dimakis
Department of Electrical and Computer Engineering
UT Austin
<EMAIL_ADDRESS>
###### Abstract

We extend Textual Inversion to learn pseudo-words that represent a concept at
different resolutions. This allows us to generate images that use the concept
with different levels of detail and also to manipulate different resolutions
using language. Once learned, the user can generate images at different levels
of agreement to the original concept; “A photo of $S^{*}(0)$” produces the
exact object while the prompt “A photo of $S^{*}(0.8)$” only matches the rough
outlines and colors. Our framework allows us to generate images that use
different resolutions of an image (e.g. details, textures, styles) as separate
pseudo-words that can be composed in various ways. We open-soure our code in
the following URL: https://github.com/giannisdaras/multires_textual_inversion.

##  1 Introduction

Textual Inversion [1] is a novel technique for introducing a new concept in a
pre-trained text conditional generative model. Given a few images of a user-
provided concept (e.g. an object or style), Textual inversion learns a new
“word" in the embedding space to represent that object. Remarkably, these new
pseudo-words can be composed in language to produce all kinds of creative
compositions.

We extend Textual inversion to learn multiple pseudo-words that represent a
concept at different resolutions. For example, in Fig. 1(a) the input concept
is the re-creation, with various small objects, of Vermeer’s Girl with a Pearl
Earring, by artist J. Perkins [2]. We learn this concept using 4 re-croppings
of the input. We then use a pre-trained text-to-image model to generate images
using prompts that contain the learned pseudo-words that represent the concept
at various resolutions.

Inputs

(a) *

A painting of a dog in the style of <jane($t_{\mathrm{fixed}}$)>.

(b) * (c) * (d) $t_{\mathrm{fixed}}=0.0$ (e) $t_{\mathrm{fixed}}=0.5$ (f)
$t_{\mathrm{fixed}}=0.7$ Figure 1: Multi-resolution textual inversion. We
learn with our method the input concept as <jane>. We then show images
generated with the prompt: A painting of a dog in the style of
<jane($t_{\mathrm{fixed}}$)>, where $t_{\mathrm{fixed}}$ controls the
resolution in which the pseudo-word captures the input concept. Each pseudo-
word represents the concept with different levels of detail with the zero
index capturing the full detailed learned concept.

[0.23] [0.69] [0.23]

##  2 Method

### Background.

Text-conditional diffusion models are trained to restore a clean image, $x$,
based on a corrupted observation, $z_{t}(x)$, and a text caption, $y$,
describing the image [3, 4, 5, 6, 7]. The corruption, which is typically
additive noise, can be either in the image itself [3, 4] or in an encoding of
the image [5]. We denote with $z_{t}(x)$ the diffused observation of image $x$
at time $t$. Text-conditional models are typically trained to minimize the
objective:

$\displaystyle
J(\theta)=\mathbb{E}_{t\sim\mathcal{U}[0,1]}\mathbb{E}_{(x,y)\sim
p}\mathbb{E}_{z_{t}\sim
q_{t}(z_{t}|x,t)}||\hat{\epsilon}_{\theta}(z_{t}(x),t,c_{\theta}(y))-\epsilon(z_{0},z_{t})||^{2},$
(1)

where $p$ is the joint distribution of image-captions, $q_{t}$ is the
distribution of the diffused observations, $y$ is the text-conditioning and
$\epsilon$ is the unscaled residual to $z_{0}(x)$.

The conditioning network, $c_{\theta}(y)$, maps the tokenized text $y$ into a
latent that is used to inform the prediction of the denoiser network,
$\hat{\epsilon}_{\theta}$. The first layer of $c_{\theta}(y)$ is typically a
lookup table, $\mathrm{emb}(y)$, that maps tokens to vectors. We decompose the
conditioning network as:
$c_{\theta}(y)=\mathrm{enc}_{\theta}(\mathrm{emb}_{\theta}(y))$.

The authors of Textual Inversion [1] consider the task of finding an
embedding, $\mathrm{emb}^{*}$ for a pseudo-token $y^{*}$ that represents a
concept described by a small collection of images $X=\\{x_{1},...,x_{N}\\}$.
They do so by solving the following optimization problem:

$\displaystyle\min_{\mathrm{emb}}\mathbb{E}_{t\in\mathcal{U}[0,1]}\mathbb{E}_{x\sim\mathcal{U}_{X}}\mathbb{E}_{z_{t}\sim
q_{t}(z_{t}|x,t)}\left|\left|\hat{\epsilon}_{\theta}(z_{t}(x),t,\mathrm{enc}_{\theta}(\mathrm{emb}))-\epsilon(z_{0},z_{t})\right|\right|^{2}.$
(2)

Intuitively, Textual Inversion is optimizing for an embedding that helps the
model predict in the most effective way the residual across all the images in
the set and across all corruption levels.

### Multiresolution Textual Inversion.

The key idea of our method is that the conditioning signal can depend on the
diffusion time (i.e. the noise level). For example, if the input image to the
diffusion model is a slightly noisy image of a cat, the text conditioning
should give information on the details of the cat (e.g. texture) and not
necessarily on the image class which can easily be inferred from the image
input. Similarly, for very noisy images the conditioning should be capturing
basic information such as image class (e.g. cat vs dog) and colors. In
general, we propose the following objective:

$\displaystyle\min_{\\{\mathrm{emb}_{0},...,\mathrm{emb}_{T-1}\\}}\mathbb{E}_{t\sim\mathcal{U}[0,1]}\mathbb{E}_{x\sim\mathcal{U}_{X}}\mathbb{E}_{z_{t}\sim
q_{t}(z_{t}|x,t)}\left|\left|\hat{\epsilon}_{\theta}(z_{t}(x),t,\mathrm{enc}_{\theta}(\mathrm{emb}_{\lfloor
t/T\rfloor}))-\epsilon(z_{0},z_{t})\right|\right|^{2}.$ (3)

After training, we have learned a set of embeddings,
$\mathrm{Emb}^{*}=\\{\mathrm{emb}_{0},...,\mathrm{emb}_{T-1}\\}$. Each of the
the elements of the set, captures different levels of detail. The idea of
having latents that describe the image with different amounts of agreement to
the input has been exploited to solve inverse problems with GANs [8, 9, 10].
In these works, the resolution is controlled by the index of the GAN layer in
which we invert our images. In our method the agreement to the input is
determined by the diffusion time index and we can manipulate the different
resolutions using language.

We note that our method can also be extended to support DreamBooth [11], an
alternative method for Textual Inversion that finetunes the whole model.

We can create embeddings indexed by continuous time by using linear
interpolations of the elements of the learned set. We present three different
ways to use the learned embeddings to sample at different levels of agreement
to the concept appearing in the input images. For a visual comparison, we show
how these methods perform for a given image at different resolution levels, in
Fig. 3.

### Fixed Resolution Sampling (Method 1).

The first method fixes the conditioning to a specific embedding throughout the
sampling. It is the most similar sampling method to Textual Inversion, in the
sense that it is using a fixed embedding for all diffusion levels. The user
can control the resolution by picking one embedding from the set
$\mathrm{Emb}^{*}$. This method allows the user to visualize what is learned
at each resolution. As we explained earlier, we expect that embeddings that
correspond to time close to $t=0$, should learn details (e.g. texture) since
those are more informative to denoise a slightly noisy image than image class
for example. Embeddings closer to $t=1$ should be related to more coarse
information about the image, e.g. what is the object, what are the colors,
etc.

For the next two sampling methods, we change the conditioning based on the
sampling time. Assume that we want to generate an image at resolution
$t=t_{\mathrm{fixed}}$. We propose two sampling methods:

### Semi Resolution-Dependent Sampling (Method 2).

Semi Resolution-Dependent Sampling uses $\mathrm{emb}_{t}$ when the sampling
time is $t$ if $t\geq t_{\mathrm{fixed}}$ and no conditioning otherwise.
Essentially, this forces the model to generate images that match (in a
distributional sense) the input images at noise level $t$. This method is
particularly useful for style-transfer and creative prompts – for some
percentage of the sampling procedure the model performs unconditional
generation (with a starting point a diffused sample that matches the original
concept with noise). This idea extends the novel SDEdit paper [12] in the
sense that it allows to do guided image synthesis (but this time starting from
language tokens).

### Fully Resolution-Dependent Sampling (Method 3).

Fully Resolution-Dependent Sampling is similar to Semi Resolution-Dependent
Sampling, but instead of doing unconditional generation for
$t<t_{\mathrm{fixed}}$, it fixes the embedding to
$\mathrm{emb}_{\mathrm{fixed}}$. This still allows for variations from the
given concept (since $\mathrm{emb}_{\mathrm{fixed}}$ only resembles it in a
given resolution), but the variations are more controlled since there is no
sampling period of unconditional generation.

### Creating a textual interface.

Similar to Textual Inversion, we create pseudo-words that allow us to use
language to guide the image generation process with our learned embeddings.
The difference is that the embeddings for our pseudo-words might be changing
based on the sampling time. We use
$S^{*}|t_{\mathrm{fixed}}|,S^{*}(t_{\mathrm{fixed}}),S^{*}[t_{\mathrm{fixed}}]$
to denote tokens that are used with Fixed Resolution, Semi Resolution and
Fully Resolution Dependent Sampling respectively.

##  3 Experiments

(a) *

(b) *

(c) *

(d) *

(e) *

(f) Inputs learned as <jane> and <cat>.

(g) A painting of a dog made with <jane$|0.1|$>.

(h) A photo of <cat$(5)$> made with <jane$|0.1|$>.

(i) A photo of <jane$(3)$> similar to <cat$[0]$>.

Figure 2: Compositions of learned pseudo-words across resolutions. Observe
that the dog (b) and cat (c) are made from small plastic objects, i.e. using
the detailed structure of <jane>, but in the shape of a dog or the toy cat. In
(d) the color stripes are obtained from the cat detail.

We begin by comparing the different sampling methods. Results are shown in
Fig. 3. Each row shows a different sampling algorithm each each column a
different resolution. With Fixed Resolution Sampling (row 1), we can
selectively extract details of the image, e.g. $S^{*}_{0.0}$ extracts only the
buttons. Semi Resolution-Dependent Sampling (row 2) allows for larger
variations from the input for high values of $t_{\mathrm{fixed}}$, e.g.
$S^{*}(0.8)$ gives a photorealistic image of person that only roughly
preserves colors and pose from the input concept. Finally, Fully Resolution-
Dependent Sampling (row 3) maintains higher agreement to the input concept
along all resolutions.

Inputs

(a) *

A photo of $S^{*}|{t_{\mathrm{fixed}}}|$ (Fixed Resolution Sampling)

(b) *

[0.15] [0.45]

(c) *

(d) *

(e) *

(f) *

(g) *

A photo of $S^{*}({t_{\mathrm{fixed}}})$ (Semi Resolution-Dependent Sampling)

(h) *

[0.15] [0.45]

(i) * (j) * (k) * (l) *

[0.15]

(m) *

A photo of $S^{*}[{t_{\mathrm{fixed}}}]$ (Fully Resolution-Dependent Sampling)

(n) *

[0.15] [0.45]

(o) * (p) $t_{\mathrm{fixed}}=0.8$ (q) $t_{\mathrm{fixed}}=0.5$ (r)
$t_{\mathrm{fixed}}=0.0$

[0.15]

Figure 3: Comparison of different sampling methods that be used for
Multiresolution Textual Inversion. Each row shows a different sampling
algorithm and each column a different resolution. With Fixed Resolution
Sampling (row 1), we can selectively extract details of the image, e.g.
$S^{*}|0.0|$ extracts only the buttons. Semi Resolution-Dependent Sampling
(row 2) allows for larger variations from the input for high values of
$t_{\mathrm{fixed}}$. Fully Resolution-Dependent Sampling (row 3) maintains
higher agreement to the inputs along all resolutions.

Inputs

Ours

Textual Inversion

(a)  (b) Watercolor painting of $S^{*}$ on a branch.

(c) Grainy photo of $S^{*}$ in angry birds. (d) Watercolor painting of $S^{*}$
on a branch.

(e) Grainy photo of $S^{*}$ in angry birds. (f)

(g) A photo of $S^{*}$ on the beach. (h) A photo of $S^{*}$ on the moon.

(i) A photo of $S^{*}$ on the beach. (j) A photo of $S^{*}$ on the moon.

(k)  (l) A photo of $S^{*}$ on the beach.

(m) A photo of $S^{*}$ on the moon. (n) A photo of $S^{*}$ on the beach.

(o) A photo of $S^{*}$ on the moon.

Figure 4: Personalized image generation. We show that with our method we can
use the pseudo-words for the learned concept to create personalized images as
if it was a normal word token. Our method performs on par and sometimes better
than Textual Inversion on this task.

We then show that Multiresolution Textual Inversion performs on par (or even
better) with Textual Inversion. Fig. 4 shows that our method can use the
learned concepts in combinations with prompts of all sorts – we use the prompt
and images from the Textual Inversion paper.

Finally, Fig. 2 shows that the learned pseudo-words can be combined in
arbitrary ways across different resolutions and concepts. For example, we can
extract the plastic object detailed structure of concept <jane> and use it to
generate other objects, such as a dog painting and the cat of Fig. 2(f).

##  4 Conclusions and Future Work

We showed how to learn multiple pseudo-words representing an input concept at
different scales. This expands the prompt vocabulary and can be easily used
with pre-trained text conditional diffusion models. As future work, we would
like to quantitatively measure if our method outperforms Textual Inversion and
if models trained with diffusion processes beyond additive noise [13, 14] can
extract different aspects of scale, depending on how they corrupt their
inputs.

## Acknowledgments and Disclosure of Funding

This research has been supported by NSF Grants CCF 1763702, AF 1901292, CNS
2148141, Tripods CCF 1934932, IFML CCF 2019844, the Texas Advanced Computing
Center (TACC) and research gifts by Western Digital, WNCG IAP, UT Austin
Machine Learning Lab (MLL), Cisco, Onassis Fellowship, Bodossaki Fellowship
and the Archie Straiton Endowed Faculty Fellowship.

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