MNRAS
510,
110–130
(2022)
https://doi.org/10.1093/mnras/stab3377
Advance
Access
publication
2021
No
v
ember
26
The
acoustic
resonant
drag
instability
with
a
spectrum
of
grain
sizes
Jonathan
Squire
,
1
‹
Stefania
Moroianu
2
and
Philip
F.
Hopkins
3
1
Physics
Department,
University
of
Otago,
Dunedin
9010,
New
Zealand
2
Department
of
Applied
Physics,
Stanford
University,
Stanford,
CA
94305,
USA
3
TAPIR,
Mailcode
350-17,
California
Institute
of
Technology,
Pasadena,
CA
91125,
USA
Accepted
2021
No
v
ember
16.
Received
2021
November
15;
in
original
form
2021
June
29
A
B
S
T
R
A
C
T
We
study
the
linear
growth
and
non-linear
saturation
of
the
‘acoustic
Resonant
Drag
Instability’
(RDI)
when
the
dust
grains,
which
drive
the
instability,
have
a
wide,
continuous
spectrum
of
different
sizes.
This
physics
is
generally
applicable
to
dusty
winds
driven
by
radiation
pressure,
such
as
occurs
around
red-giant
stars,
star-forming
regions,
or
active
galactic
nuclei.
Depending
on
the
physical
size
of
the
grains
compared
to
the
wavelength
of
the
radiation
field
that
drives
the
wind,
two
qualitatively
different
regimes
emerge.
In
the
case
of
grains
that
are
larger
than
the
radiation’s
wavelength
–termed
the
constant-drift
regime
–the
grain’s
equilibrium
drift
velocity
through
the
gas
is
approximately
independent
of
grain
size,
leading
to
strong
correlations
between
differently
sized
grains
that
persist
well
into
the
saturated
non-linear
turbulence.
For
grains
that
are
smaller
than
the
radiation’s
wavelength
–termed
the
non-constant-drift
regime
–the
linear
instability
grows
more
slowly
than
the
single-grain-
size
RDI
and
only
the
larger
grains
exhibit
RDI-like
behaviour
in
the
saturated
state.
A
detailed
study
of
grain
clumping
and
grain–grain
collisions
shows
that
outflows
in
the
constant-drift
regime
may
be
ef
fecti
ve
sites
for
grain
growth
through
collisions,
with
large
collision
rates
but
low
collision
velocities.
Key
words:
instabilities
– turbulence
–stars:
winds,
outflows
– dust,
extinction
–ISM:
kinematics
and
dynamics
– galaxies:
formation.
1
INTRODUCTION
Cosmic
dust
is
ubiquitous
across
the
Universe
and
vital
to
a
wide
range
of
astrophysical
processes.
By
mass,
it
makes
up
around
∼
1
per
cent
of
the
interstellar
medium
(ISM)
of
galaxies,
but
its
strong
coupling
to
radiation
fields
implies
it
can
none
the
less
strongly
influence
gas
dynamics
and
cooling
in
many
situations
(Draine
2010
).
More
generally,
because
around
half
of
the
metal
content
of
our
Galaxy
is
locked
up
in
dust,
it
plays
crucial
roles
in
any
process
that
requires
metals
or
solids
(Whittet
1992
;
Draine
2003
).
Notably,
dust
is
almost
certainly
the
key
ingredient
for
planet
formation
and
life,
supplying
the
necessary
reservoir
of
solids
that
provide
the
seeds
to
make
planetesimals
in
protostellar
discs
(Chiang
&
Youdin
2010
).
This
paper
deals
with
the
physics
of
dust
moving
through
gas,
with
the
interaction
between
the
species
mediated
by
drag
forces.
Such
conditions
occur,
for
example,
in
dust-radiation-pressure
driven
winds,
where
an
outflow
of
dusty
gas
is
driven
by
an
anisotropic
radiation
field
that
couples
strongly
to
the
dust.
Such
outflows
are
thought
to
be
important
in
the
evolution
of
asymptotic
giant
branch
(AGB)
stars
(which
also
produce
large
quantities
of
dust;
e.g.
Habing
1996
;
Norris
et
al.
2012
),
in
feedback
processes
that
help
regulate
star
formation
and/or
active
galactic
nuclei
(AGNs;
Scoville
2003
;
Thompson,
Quataert
&
Murray
2005
;
Ishibashi
&
Fabian
2015
),
around
supernovae
(e.g.
Micelotta,
Matsuura
&
Sarangi
2018
),
and
in
the
bulk
ISM
(Weingartner
&
Draine
2001b
)
and
circumgalactic
medium
(CGM;
M
́
enard
et
al.
2010
).
As
shown
by
Squire
&
Hopkins
E-mail:
jonathan.squire@otago.ac.nz
(
2018a
),
this
situation
– specifically,
when
the
radiation
pressure
on
the
dust
is
stronger
than
that
on
the
gas,
such
that
the
gas
outflow
is
driven
indirectly
through
the
drag
force
from
the
dust
–is
unstable
to
the
‘Resonant
Drag
Instability’
(RDI):
small
perturbations
in
the
gas
or
dust
will
grow
in
time
until
they
become
large,
driving
turbulence
in
the
gas
and
strong
dust
clumping.
Hopkins
&
Squire
(
2018a
)
(hereafter
HS18
)
studied
the
linear
features
and
growth
rates
of
the
RDI
for
the
case
of
nearly
neutral
grains
and
neutral
gas
in
outflows,
while
Hopkins
&
Squire
(
2018b
)
generalized
these
results
to
charged
grains
in
magnetized
gas.
These
results
were
then
extended
to
the
non-linear
regime
by
Moseley,
Squire
&
Hopkins
(
2019
)
(hereafter
MSH19
)
and
Seligman,
Hopkins
&
Squire
2019
;
Hopkins,
Squire
&
Seligman
2020
(in
the
magnetized
regime),
who
studied
the
turbulence
induced
by
the
RDI,
constructing
simple
estimates
for
its
saturation
amplitude
and
other
properties.
Ho
we
ver,
each
of
these
studies
has
allowed
for
the
dust
grains
to
have
only
a
single
size.
As
shown
by
Krapp
et
al.
(
2019
),
P
aardekooper,
McNally
&
Lo
vascio
(
2020
),
and
Zhu
&
Ya n g
(
2021
)
for
the
streaming
instability
in
protoplanetary
discs
(which
is
part
of
the
RDI
family;
Squire
&
Hopkins
2018b
),
a
range
of
grain
sizes
can
strongly
influence
the
instabilities
in
some
regimes.
Thus,
for
more
realistic
application
to
astrophysical
fluids
and
outflows,
where
the
range
in
grain
sizes
easily
spans
two
orders
of
magnitude
or
more
(Draine
2010
),
we
must
relax
the
single-grain-size
assumption
and
better
understand
the
influence
of
a
spectrum
of
grain
sizes
on
the
growth
rate
and
saturation
of
the
RDI.
This
is
the
first
purpose
of
this
paper.
We
study
both
the
linear
growth
rates
and
non-linear
saturation
of
the
‘acoustic
RDI’
(
HS18
)
that
applies
to
uncharged
grains
and
neutral
gas,
and
involves
the
driving
of
compressive
shocks
and
© 2021
The
Author(s)
Published
by
Oxford
University
Press
on
behalf
of
Royal
Astronomical
Society
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The
RDI
with
a
spectrum
of
grains
111
sound
waves
by
the
drifting
dust.
We
find
that
depending
on
whether
grains
are
smaller
or
larger
than
the
wavelength
of
the
accelerating
radiation
field,
the
presence
of
a
spectrum
of
grains
either
has
little
effect
on
the
acoustic
RDI
or
reduces
the
growth
rate
and
saturation
of
smaller
scale
motions.
In
both
cases,
key
features
of
the
linear-
instability
structure
persist
well
into
the
highly
turbulent
saturated
state.
The
second
purpose
of
this
paper
is
to
better
understand
dust
clumping
and
collisions
in
RDI
turbulence
(i.e.
the
saturated
state
of
the
acoustic
RDI).
This
is
important
because
outflows
are
highly
dynamic
and
often
thought
to
be
key
sites
for
grain
condensation,
coagulation,
and
fragmentation,
and
the
latter
two
of
these
processes
are
strongly
influenced
by
turbulence.
While
well-developed
theories
exist
to
describe
the
how
standard
gas
turbulent
motions
influence
dust
collisions
(e.g.
Ormel
&
Cuzzi
2007
;
Pan
&
Padoan
2013
;
Pumir
&
Wilkinson
2016
),
the
structure
of
the
turbulence
and
clumping
driven
by
the
acoustic
RDI
is
quite
different
to
standard
turbulence
in
many
ways,
because
the
instability
operates
across
all
scales
of
the
system
simultaneously
(
MSH19
).
In
this
context,
it
is
important
to
consider
the
RDI
with
a
wide
spectrum
of
grain
sizes
(as
opposed
to
the
single-grain-size
RDI)
because
the
nature
of
the
instability
suggests
that
there
could
exist
interesting
correlations
between
differently
sized
grains
in
some
regimes,
and
grain
clumping
and
collision
statistics
depend
strongly
on
grain
sizes
(P
an,
P
adoan
&
Scalo
2014b
;
Blum
2018
;
Mattsson
et
al.
2019
).
With
this
in
mind,
our
non-linear
study
is
designed
to
compare
RDI
saturation
to
forced
turbulence
with
passively
advected
dust.
We
do
this
by
designing
‘equi
v
alent’
forced
turbulence
simulations
(i.e.
simulations
with
parameters
chosen
to
match
the
saturated
RDI
as
closely
as
possible),
allowing
an
explicit
comparison
of
the
statistics
of
dust
in
RDI
turbulence
with
those
of
passive
dust
in
forced
turbulence.
Given
the
rather
detailed
nature
of
these
comparisons,
we
focus
on
just
two
RDI
case
studies
at
the
high
numerical
resolution,
but
with
parameters
that
can
be
applicable
to
a
range
of
astrophysical
scenarios.
Depending
on
the
regime,
we
find
that
the
RDI
involves
a
significantly
faster
rate
of
lower
velocity
collisions
than
forced
turbulence,
particularly
between
grains
of
different
sizes.
It
also
exhibits
far
stronger
clumping
of
the
smallest
grains,
even
when
simple
estimates
suggest
these
small
grains
should
be
very
well
coupled
to
the
gas.
The
paper
is
split
in
two:
first
the
main
exposition;
second
an
extended
appendix
that
studies
analytically
the
linear
behaviour
of
the
acoustic
RDI
with
a
spectrum
of
grain
sizes.
This
split
was
chosen
because
the
linear
calculations,
in
which
we
derive
simple
analytic
expressions
for
the
RDI
growth
rate
in
most
of
the
important
regimes,
are
necessarily
rather
technical.
They
do
provide
useful
understanding
of
the
non-linear
behaviour,
ho
we
ver,
so
we
will
refer
to
those
results
throughout.
The
main
paper
starts
in
Section
2
with
a
detailed
description
of
the
problem,
model,
and
numerical
setup,
particularly
focusing
on
important
differences
that
arise
in
the
dust
(quasi-)equilibrium
depending
on
the
wavelength
of
the
radiation
compared
to
the
dust
size
(Section
2.2
and
Ta b l e
1
).
The
numerical
re-
sults
are
presented
in
Section
3,
starting
with
a
discussion
of
the
broad
morphological
features
of
the
turbulence
and
how
this
differs
between
regimes
and
driving
(Section
3.1),
then
followed
by
a
more
detailed
analysis
of
the
grain
clumping
and
collisions
(Sections
3.2
and
3.3).
2
NUMERICAL
MODEL
AND
PHYSICAL
SETUP
2.1
Dust
and
gas
model
We
model
gas
dynamics
using
the
standard
neutral
fluid
equations,
ignoring
magnetohydrodynamic
effects
and
charged
grains
for
sim-
plicity
in
this
study,
although
such
effects
are
important
for
many
astrophysical
regimes
(
HS18
).
Dust
is
modelled
numerically
by
treating
it
as
a
population
of
individual
particles
(the
superparticle
approach),
which
interact
with
the
gas
through
drag
forces
that
depend
on
the
grain
size.
We
use
f
d
(
grain
;
x
,
v
,
t
)
to
denote
the
phase-
space
density
of
grains
with
radius
grain
,
velocity
v
,
at
position
x
,
such
that
the
equations
of
motion
are
∂ρ
g
∂t
+
∇
·
(
ρ
g
u
g
)
=
0
,
(1)
∂
u
g
∂t
+
u
g
·∇
u
g
=
−
c
2
s
∇
ρ
g
ρ
g
+
1
ρ
g
d
grain
d
v
f
d
(
x
,
v
,
grain
)
v
−
u
g
t
s
(
grain
,
v
)
,
(2)
∂f
d
∂t
+
v
·∇
f
d
+
∂
∂
v
·
−
v
−
u
g
(
x
)
t
s
(
grain
,
v
)
+
a
ext
(
grain
)
f
d
=
0
.
(3)
Given
the
superparticle
approach,
the
final
equation
is
equi
v
alent
to
modelling
individual
grains
of
size
grain,
j
,
with
velocity
v
j
and
position
x
j
,
with
∂
x
j
∂t
=
v
j
,
∂
v
j
∂t
=
−
v
j
−
u
g
(
x
j
)
t
s
(
grain
,j
,
v
j
)
+
a
ext
(
grain
,j
)
(4)
as
the
y
mo
v
e
through
the
gas
velocity
field,
then
constructing
f
d
by
counting
dust
particles
at
each
gas
position.
Here,
ρ
g
is
the
gas
density,
u
g
is
the
gas
velocity,
c
s
is
the
sound
speed
(the
equation
of
state
is
taken
as
isothermal),
t
s
(
grain
,
v
)
is
the
stopping
time
(see
equation
7
below),
and
a
ext
is
an
external
force
from
radiation
pressure
on
the
grains,
which
we
will
take
(arbitrarily)
to
be
in
the
z
direction
a
ext
=
a
ext
z
(see
Section
2.2).
The
final
term
in
equation
(2)
is
the
dust
‘backreaction’
force
–it
the
force
on
the
gas
from
the
dust
–
which
is
neglected
in
most
studies
of
dust
dynamics.
We
also
use
·
to
denote
the
v
olume
a
verage,
and
the
subscripts
⊥
and
to
denote
velocities
in
the
directions
perpendicular
and
parallel
to
the
mean
drift,
respectively
(e.g.
u
2
g
,
=
u
2
g
,z
,
u
2
g
,
⊥
=
u
2
g
,x
+
u
2
g
,y
).
A
δ
indicates
that
the
mean
is
subtracted
from
a
quantity
before
averaging
(e.g.
δv
d,
=
v
d,
−
v
d,
).
All
quantities
will
be
measured
in
the
frame
where
the
gas
is
stationary
u
g
=
0.
To
understand
dust
dynamics,
analyse
simulations,
and
compute
linear
growth
rates,
it
is
helpful
to
take
the
zeroth
and
first
velocity
moments
of
f
d
,
defining
ρ
d
=
gr
,h
gr
,l
d
grain
d
v
f
d
,
v
d
=
1
ρ
d
gr
,h
gr
,l
d
grain
d
v
v
f
d
,
(5)
where
the
integration
limits
gr,
l
and
gr,
h
can
be
taken
over
just
a
subset
of
grains
(if
specified
as
such;
see
Section
2.4.2),
or
the
full
distribution
(if
unspecified).
We
use
μ
0
to
denote
the
total
average
dust-to-gas-mass
ratio
μ
0
=
ρ
d
/
ρ
g
,
and
w
s
(
grain
)
=
|
v
d
(
grain
)
−
u
g
|
c
s
,
(6)
to
denote
the
mean
drift
between
dust
of
size
grain
and
the
gas
(here
v
d
(
grain
)
is
computed
from
gr,
l
=
grain
,
gr,
h
=
grain
+
d
grain
in
equation
5).
Throughout,
we
assume
Epstein
drag,
using
the
simple
approxi-
mation
of
(Draine
&
Salpeter
1979
),
t
s
(
grain
,
v
)
=
π
8
̄
ρ
d
,
int
grain
ρ
g
c
s
1
+
9
π
128
|
v
−
u
g
|
2
c
2
s
−
1
/
2
,
(7)
where
̄
ρ
d
,
int
is
the
internal
grain
density.
Epstein
drag
is
generally
appropriate
for
astrophysical
conditions
in
which
MHD
and
charging
MNRAS
510,
110–130
(2022)
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112
J.
Squire,
S.
Moroianu,
and
P.
F.
Hopkins
Table
1.
Summary
of
some
basic
properties
of
the
grains
and
the
RDI
in
the
constant-
and
non-constant-drift
regimes.
Constant
drift
regime
Non-constant
drift
regime
λ
rad
<
grain
,
Q
abs
∼
1
(radiative
force
on
the
dust)
λ
rad
>
grain
,
Q
abs
∼
grain
/
λ
rad
(force
on
the
dust)
or
acceleration
of
the
gas
a
ext
∝
−
1
grain
a
ext
∼
const.
w
s
∼
const.,
t
s
∝
grain
w
s
∝
1
/
2
grain
,
t
s
∝
1
/
2
grain
(supersonic
drift);
w
s
∝
grain
,
t
s
∝
grain
(subsonic
drift)
Linear
instability
similar
to
single-grain-size
case
Range
of
resonant
angles
changes
character
of
linear
instability
Strong
correlations
between
grains
of
different
sizes
in
saturated
state
Grain
correlations
broadly
similar
to
externally
forced
turbulence
effects
can
be
neglected
as
done
here
(generally,
for
cooler,
denser
gas;
see
e.g.
HS18
).
2.1.1
Grain
mass
distribution
We
will
assume
a
simple
power-law
distribution
of
grain
sizes
be-
tween
grain
=
min
grain
and
grain
=
max
grain
.
In
order
to
reduce
the
number
of
free
parameters,
we
use
the
standard
MRN
distribution
(Mathis,
Rumpl
&
Nordsieck
1977
),
which
postulates
that
the
mass
of
grains
d
μ
within
logarithmic
range
of
sizes
d
ln
grain
is
d
μ/
d
ln
grain
∝
0
.
5
grain
,
such
that
most
of
the
mass
is
in
the
largest
grains,
along
with
the
total
dust-to-gas-mass
ratio
μ
0
≈
0.01.
The
original
distribution
postulates
that
min
grain
≈
5
nm
and
max
grain
≈
0
.
25
μ
m
,
although
subse-
quent
works
have
suggested
that
this
underestimates
significantly
the
population
of
small
grains
and
misses
a
population
of
larger
grains,
even
in
the
diffuse
ISM
(Weingartner
&
Draine
2001a
;
Zubko,
Dwek
&
Arendt
2004
;
Draine
&
Fraisse
2009
).
Less
is
known
about
the
grain
distribution
in
more
dynamic
environments
(e.g.
around
AGB
stars
or
the
AGN
dusty
torus,
see
e.g.
H
̈
ofner
&
Olofsson
2018
;
Murray,
Quataert
&
Thompson
2005
),
where
our
simulations
can
apply
by
virtue
of
their
dimensionless
nature
(see
Section
2.5
below).
Cursory
lower
resolution
tests
of
different
grain-mass
distributions
(e.g.
small-grain
dominated,
d
μ/
d
ln
grain
∝
−
0
.
5
grain
)
have
not
revealed
significant
differences,
so
we
shall
not
explore
this
in
detail.
It
is,
ho
we
ver,
worth
noting
that
for
the
gas
of
the
protoplanetary-
disc
streaming
instability,
the
grain
distribution
can
affect
important
details
of
the
linear
instability
(Paardekooper
et
al.
2020
;
McNally,
Lovascio
&
Paardekooper
2021
;
Zhu
&
Ya n g
2021
),
so
this
issue
may
be
worth
revisiting
in
more
detail
in
future
work.
2.2
Grain
acceleration
regimes
The
acoustic
RDI
requires
a
net
drift
between
grains
and
the
gas
as
their
energy
source.
As
discussed
e
xtensiv
ely
in
HS18
and
Hopkins
&
Squire
(
2018b
),
such
a
net
drift
is
expected
to
occur
generically
in
the
presence
of
radiation
fields,
which
couple
to
the
grains
more
strongly
than
to
the
gas
under
most
conditions
(e.g.
Weingartner
&
Draine
2001b
;
Murray
et
al.
2005
),
sourcing
the
a
ext
term
in
equation
(3).
In
this
radiatively
driven
situation,
two
acceleration
regimes
naturally
emerge,
applying,
respectively,
to
grains
smaller
or
larger
than
the
wavelength
of
the
accelerating
radiation
λ
rad
.
The
different
scaling
of
w
s
(
grain
)
has
important
implications
for
the
development
of
instabilities.
Some
basic
properties
of
the
different
regimes
are
summarized
in
Ta b l e
1
for
quick
reference.
A
coherent
radiation
field
of
energy
density
e
rad
causes
a
grain
acceleration
a
ext
∼
Q
abs
e
rad
2
grain
/m
d
c
2
,
where
m
d
=
4
/
3
π
̄
ρ
d
3
grain
is
the
grain’s
mass.
The
factor
Q
abs
is
the
absorption
efficiency;
Q
abs
∼
1
if
grain
λ
rad
,
while
Q
abs
∼
grain
/
λ
rad
if
grain
λ
rad
(Weingartner
&
Draine
2001b
).
Thus,
large
grains
in
shorter
wavelength
radiation
fields
feel
an
acceleration
a
ext
∝
−
1
grain
,
while
small
grains
in
longer
wavelength
radiation
fields
feel
an
acceleration
that
is
independent
of
grain
.
In
such
a
driven
situation,
the
(quasi-)equilibrium
occurs
when
the
acceleration
and
drag
forces
on
grains
balance,
1
viz.,
when
w
s
(
grain
,
v
d
)
t
s
(
grain
,
v
d
)
=
a
ext
(
grain
)
.
(8)
Because
t
s
(
grain
,
v
d
)
∝
grain
,
this
implies
that
w
s
is
independent
of
grain
size
if
grain
λ
rad
,
which
we
term
the
constant
drift
regime;
the
opposite
regime,
where
w
s
(
grain
)
is
a
function
of
grain
size
(for
grain
λ
rad
)
is
termed
the
non-constant
drift
regime.
In
the
latter
case,
we
see
from
equations
(7)
and
(8)
that
for
subsonic
drift
w
s
1,
t
s
in
the
saturated
state
is
independent
of
w
s
so
that
t
s
∝
grain
and
w
s
∝
grain
,
while
for
supersonic
drift
w
s
1
(the
case
of
more
rele
v
ance
to
this
article),
the
decrease
in
t
s
with
w
s
implies
that
t
s
∝
1
/
2
grain
and
w
s
∝
1
/
2
grain
.
Of
course,
in
many
physical
situations,
the
distribution
of
grain
sizes
could
fall
around
λ
rad
(i.e.
min
grain
<
λ
rad
<
max
grain
),
in
which
case
the
larger
grains
will
lie
in
the
constant-drift
regime
and
the
smaller
grains
in
the
non-constant-drift
regime.
Ho
we
ver,
gi
ven
that
our
goal
here
is
to
explore
the
basic
physics
of
the
multigrain
RDI,
we
will
not
consider
such
situations
in
detail
in
this
work.
A
net
drift
of
grains
can
also
be
set
up
through
a
force
on
the
gas
(and
not
the
dust),
due
to
radiation
pressure
absorbed
by
gas
(e.g.
line
pressure)
or
gravity
(e.g.
in
a
stratified
medium).
In
this
case,
the
situation
is
identical
to
the
non-constant-drift
regime.
This
physics
is
applicable,
albeit
with
a
different
linear
instability,
to
the
polydisperse
streaming
instability
in
protoplanetary
discs
(Krapp
et
al.
2019
).
Note
that
Hopkins
et
al.
(
2021
)
also
explore
the
RDI
with
a
spectrum
of
grain
sizes,
explicitly
including
vertical
stratification
and
more
complex
radiation-MHD
effects.
For
some
cases,
they
also
consider
simulation
variants
in
the
grain
λ
rad
and
grain
λ
rad
regimes.
Rather
than
using
the
labels
‘constant-drift’
and
‘non-
constant-drift,’
they
label
the
grain
λ
rad
(non-constant-drift)
regime
simulations
with
‘
-Q
,’
to
signify
that
Q
abs
depends
on
grain
in
this
regime.
This
different
nomenclature
is
used
their
because
most
of
their
simulations
use
explicit
radiation-MHD
effects,
which
leads
to
more
complex
relations
between
w
s
and
grain
that
depend
on
the
dynamics
of
the
radiation
field.
2.3
Behaviour
of
the
acoustic
Resonant
Drag
Instability
As
discussed
in
detail
in
Squire
&
Hopkins
(
2018a
),
HS18
,
and
MSH19,
the
acoustic
RDI
exhibits
different
behaviors
based
on
a
dimensionless
scale
parameter
k
c
s
t
s
,
where
k
is
the
wavenumber
of
the
mode.
In
Appendix
A,
we
co
v
er
in
detail
the
linear
behaviour
of
the
acoustic
RDI
with
a
spectrum
of
grain
sizes,
showing,
as
1
Note
that
this
is
not
actually
a
true
equilibrium
because
it
occurs
with
the
system
as
a
whole
(gas
and
dust)
accelerating
linearly
at
a
constant
rate,
unless
a
ext
can
be
balanced
by
another
external
force
on
both
gas
and
dust
(e.g.
gravity).
Ho
we
ver,
this
subtlety
does
not
make
a
difference
to
the
arguments
here
or
to
our
simulations.
The
issue
is
discussed
in
detail
in
HS18
.
MNRAS
510,
110–130
(2022)
Downloaded from https://academic.oup.com/mnras/article/510/1/110/6442271 by Universitetsbiblioteket i Bergen user on 28 February 2022
The
RDI
with
a
spectrum
of
grains
113
expected,
that
the
same
parameter
(which
now
covers
a
range
of
values
because
of
the
range
in
grain
)
has
a
similar
influence
on
the
instability
.
Specifically
,
three
regimes
emerge:
the
instability
is
in
the
low-
k
regime
if
k
c
s
t
s
μ
;
the
mid-
k
regime
if
μ
k
c
s
t
s
μ
−
1
;
and
the
high-
k
regime
if
μ
−
1
k
c
s
t
s
.
With
a
spectrum
of
grains,
there
is
ambiguity
surrounding
these
delineations
(
t
s
and
μ
depend
on
grain
),
but
they
are
none
the
less
useful
for
general
understanding
(see
Appendix
A
for
more
precise
estimates).
In
the
low-
k
regime,
the
fastest
growing
modes
are
generally
non-resonant
and
grow
in
the
direction
aligned
with
the
drift,
involving
strong
perturbations
to
both
the
gas
and
dust
densities.
We
show
in
Section
A3.1
that
such
modes
are
generally
agnostic
to
the
presence
of
a
spectrum
of
grain
sizes
or
the
drift
regime
(constant
versus
non-constant).
In
contrast,
modes
in
the
mid-
and
high-
k
regimes
are
fastest
growing
at
a
specific
‘resonant’
mode
direction,
where
the
projection
of
the
dust
drift
speed
on
to
that
direction
is
equal
to
the
sound
speed.
The
y
hav
e
a
much
stronger
dust–density
response
than
gas–density
response.
Such
modes
behave
similarly
in
the
constant-drift
regime
with
a
spectrum
of
grain
sizes,
but
are
significantly
modified
in
the
non-constant-drift
regime
because
each
grain
size
resonates
with
a
different
mode
angle
(see
Appendix
A3).
In
addition
to
the
linear
behaviour,
the
three
regimes
control
the
acoustic
RDI’s
non-linear
evolution
(
MSH19
).
While
gas
motions
and
dust
clumping
driven
by
the
acoustic
RDI
in
the
low-
k
regime
broadly
resemble
standard
driven
supersonic
turbulence
(although
there
are
distinct
differences),
the
mid-
and
high-
k
regimes
are
very
different,
with
the
resonant
mode
structure
remaining
clear
well
into
the
saturated
state
and
across
all
scales.
In
larger-
w
s
cases,
this
manifests
itself
through
thin
dust
filaments,
which
‘draft’
on
the
nearby
dust
and
never
reach
a
saturated
turbulent
steady
state.
For
subsonic
drift,
the
linear
and
non-linear
behaviour
is
most
similar
to
the
low-
k
regime
(indeed,
the
subsonic
instability
at
mid-
to
high-
k
is
non-resonant
and
depends
on
details
of
the
equation
of
state
and
drag
law;
HS18
).
Finally,
it
is
worth
reiterating
from
previous
works
that
the
acoustic
RDI
generally
has
no
preferred
scale
in
any
of
the
three
regimes.
Rather,
modes
at
smaller
scales
grow
faster,
with
the
growth
rate
(
ω
)
scaling
as
(
ω
)
∼
k
2/3
,
(
ω
)
∼
k
1/2
,
and
(
ω
)
∼
k
1/3
,
in
the
low-
,
mid-,
and
high-
k
re
gimes,
respectiv
ely.
Thus,
simulations
cannot
be
converged
in
the
conventional
sense,
in
that
a
higher
resolution
simulation
will
resolve
faster-growing
modes
(in
the
absence
of
a
small-scale
dissipati
ve
ef
fect
such
as
viscosity).
Ho
we
ver,
as
sho
wn
by
MSH19
(appendix
B3),
the
bulk
properties
of
the
saturated
state
are
ef
fecti
vely
resolution
independent
once
box-scale
modes
saturate
non-linearly.
2.4
Simulation
design
We
use
the
code
GIZMO,
2
which
solves
the
fluid
equations
using
the
second-order
Lagrangian
‘Meshless
Finite
Volume’
(MFV)
method
(Hopkins
2015
).
Dust
is
included
via
the
super-particle
approach
(Youdin
&
Johansen
2007
;
Hopkins
&
Lee
2016
),
using
a
random
sampling
of
grain
sizes
grain
across
the
full
continuous
distribution
(i.e.
we
do
not
use
a
set
number
of
grain-size
bins).
The
backreaction
force
of
the
dust
on
the
gas
is
computed
using
a
standard
momentum-
conserving
scheme
(Youdin
&
Johansen
2007
),
with
details
of
the
scheme
and
a
variety
of
numerical
tests
described
in
appendix
B
of
MSH19
(although
MSH19
considers
only
a
single
grain
size,
there
2
A
public
version
of
the
code,
including
all
methods
used
in
this
paper,
is
available
at
ht
tp://www.tapir.calt
ech.edu/phopkins/Sit
e/GIZMO.ht
ml
are
no
significant
numerical
complications
that
arise
from
the
use
of
a
spectrum
of
grains).
The
range
of
available
parameter
space
for
simulations
using
a
grain-size
spectrum
is
e
xtensiv
e,
ev
en
without
magnetization
and
grain
charge:
it
includes
the
drift-velocity
regimes
(constant
versus
non-constant,
supersonic
versus
subsonic,
and
mixtures
of
each),
stopping-time
distributions
(which
could
straddle
the
different
k
regimes),
the
distribution
of
grain
masses,
and
the
total
dust-to-
gas-mass
ratio.
Because
of
this,
and
moti
v
ated
by
the
goal
of
better
understanding
RDI-turbulence
physics
rather
than
detailed
matching
of
specific
astrophysical
situations,
we
choose
in
this
article
to
focus
on
the
detailed
understanding
of
just
two
sets
of
RDI
parameters.
We
supplement
this
by
comparing
these
directly
to
simulations
without
dust
backreaction
(
μ
0
=
0),
where
turbulence
is
driven
by
large-
scale
external
forcing
to
have
a
similar
velocity
dispersion.
The
purpose
of
this
comparison
is
two
fold:
first,
and
most
importantly,
it
enables
us
to
probe
the
physics
of
dust
clumping
in
RDI-generated
turbulence
by
direct
comparison
to
the
better-understood
case
of
standard
(Kolmogoro
v)
turbulence,
rev
ealing
clearly
their
most
important
differences.
Secondly,
in
application
to
AGB-star
winds
or
AGN
outflows,
the
forced-turbulence
simulation
could
provide
a
reasonable
model
for
dust
clumping
if
the
RDI
were
not
operating.
Specifically,
one
might
expect
larger
scale
global
instabilities
(e.g.
Rayleigh–Taylor-like
instabilities;
Krumholz
&
Thompson
2012
)
to
drive
turbulence,
which
would
then,
through
a
turbulent
cascade,
drive
fluctuations
on
the
small
scales
considered
here.
Thus,
the
simulations
act
as
a
benchmark
for
how
such
a
dust-driven
wind
might
clump
dust
in
the
absence
of
RDIs
(e.g.
at
extremely
small
dust-to-gas
ratios),
although
we
caution
that
the
magnitude
of
the
turbulent
driving
is
not
at
all
tuned
to
explore
this
in
detail
(rather
it
is
tuned
to
address
the
first
point
and
probe
the
physics).
The
o
v
erall
approach
complements
and
builds
on
that
of
MSH19
and
Hopkins
et
al.
(
2020
),
which
surv
e
yed
a
wide
range
of
parameters
to
understand
how
the
RDI
behaves
in
different
regimes.
Specifically,
the
results
of
MSH19
tell
us
that
the
most
interesting
computationally
accessible
RDI
behaviour
–i.e.
that
which
exhibits
the
most
inter-
esting
differences
compared
to
standard
turbulence
–occurs
in
the
‘mid-
k
’
range.
As
discussed
abo
v
e
(Section
2.3)
this
regime
is
also
expected
to
show
more
interesting
differences
between
the
grain-
spectrum
and
single-grain
RDIs,
so
is,
o
v
erall,
the
most
obvious
candidate
for
further
study.
3
Thus
we
will
ignore
the
low-
k
regime
and/or
subsonic
drift
in
this
study.
Although
they
are
potentially
astrophysically
rele
v
ant
in
many
situations
(Hopkins
et
al.
2021
),
these
regimes
can
likely
be
mostly
adequately
understood
using
a
combination
of
RDI-related
understanding
from
MSH19
and
HS18
and
theories
of
collisions/clustering
in
turbulence
with
a
spectrum
of
sizes
without
dust
backreaction
(e.g.
Pan
et
al.
2014b
;
Pan
&
Padoan
2014
;
Hopkins
&
Lee
2016
;
Mattsson
et
al.
2019
;
Li
&
Mattsson
2020
).
Based
on
the
discussion
of
the
previous
paragraph,
we
focus
on
four
simulations
with
a
grain-size
spectrum
co
v
ering
a
factor
of
100
(
max
grain
=
100
min
grain
),
each
in
a
cubic
box
of
size
L
3
.
These
are:
Constant-drift
RDI
:
This
simulation
sets
a
ext
∝
1/
grain
such
that
w
s
is
independent
of
grain
and
t
s
∝
grain
.
The
acceleration
and
3
Unfortunately,
reaching
the
true
high-
k
regime
has
proved
to
be
computa-
tionally
challenging,
because
the
width
of
the
resonant
wavelengths
become
increasingly
narrow
at
increasing
k
,
necessitating
e
xcessiv
ely
high
resolution.
MSH19
considered
some
cases
around
the
boundary
between
the
mid-
and
high-
k
regimes,
which
were
similar
to
the
mid-
k
cases
(see
their
appendix
A
for
more
information).
MNRAS
510,
110–130
(2022)
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