potential (e.g. Daniel & Wyse, 2018). Also Solway et al. (2012) showed, and
Kordopatis et al. (2015) found supporting empirical evidence, that migration
is only mildly reduced by vertical motion. This is because the vertical
excursions of even thick disk stars are a small fraction of an open, low-$m$
spiral’s crest-to-crest wavelength, $\lambda_{\perp}=2\pi R\sin\alpha/m$,
where $\alpha$ is the pitch angle of the spiral. Note that the potential of a
WKB wave (eq. 1) decays away from the disk mid-plane as
$\exp(-2\pi|z|/\lambda_{\perp})$.

####  7.1.2 Random Walk in Radius

Changes to the guiding center radii caused by a series of transient spiral
modes with corotation radii scattered over a wide swath of the disk will cause
some stars to execute a random walk in radius, while preserving radial and
vertical actions (e.g. Mikkola et al., 2020). Typical step sizes range to over
$\sim 2\;$kpc (Sellwood & Binney, 2002, Roškar et al., 2012, Aumer et al.,
2016), though they are smaller, and consequently cause weaker churning
(Schönrich & Binney, 2009), for lower amplitude spirals, such as occur in
simulations having hotter and thicker disks (see §7.5). The resulting churning
of the stellar distribution has implications for abundance gradients and age-
metallicity relations. ChurningThe shuffling of the angular momenta of disk
stars by radial migration

Minchev et al. (2012) suggest that bars play a role in radial mixing, which
they argue is enhanced by overlap between the resonances of the bar and
spirals. Note, however, that bars themselves tend not to be transient
disturbances, and therefore stars that are trapped in a CR with the bar will
repeatedly cross and recross that resonance. Indeed, the trapping of stars
near corotation of the bar could be important for the maintenance of inner
rings (Buta & Combes, 1996). An essential aspect of radial migration by
spirals is that the pattern has decayed before there is time for the star to
make a second crossing of the CR of a spiral, leading to a lasting change in
$L_{z}$. Resonance overlap could perhaps provide a route for particles trapped
in the CR of the bar to escape to the outer disk.

The underlying metallicity gradients are also blurred by epicyclic motions.
Since the guiding center radius of a star is determined only by its angular
momentum, the intrinsic radial gradient without blurring can be estimated from
samples of Milky Way stars without having to integrate their orbits.
BlurringThe smearing effect caused by epicyclic motions of the stars that may
reduce an underlying radial metallicity gradient

####  7.1.3 Radial Migration in Sub-maximum Disks

Simulations of sub-maximum disks support multi-arm patterns (as noted in
§5.2.2) and therefore differ from the low-order rotational symmetry preferred
in galaxies (§2.2). The crest-to-crest wavelength
$\lambda_{\perp}=2\pi/k_{\perp}$ of such multi-arm patterns is generally much
shorter than that in all but the most tightly wrapped $m=2$ spirals, which has
several consequences that reduce radial migration.

  1. 1.

For a fixed density amplitude, $\Sigma_{a}$ in eq. (1), the spiral potential
is weaker for larger $k_{\perp}$, which will diminish the radial extent of the
horseshoe orbit region that is responsible for migration.

  2. 2.

The period of a horseshoe orbit trapped in the CR depends on both its
frequency difference from CR and the azimuthal distance between wave-crests,
which is shorter for higher-$m$ spirals. Since migration relies on the spiral
having already decayed before a star makes its second horseshoe turn, those
stars having periods long-enough to make only a single turn are confined to a
narrower region about CR, implying a smaller average step size for migration.

  3. 3.

The increased value of $k_{\perp}$ causes the spiral potential (eq. 1) to
decay away from the mid-plane more rapidly, lessening its ability to affect
the orbits of thick disk stars.

These factors, which stem from the short wavelength of the spirals, will
reduce the extent of churning that is possible in both the thin and thick
disks in simulations of atypically sub-maximum disks, as reported by Vera-Ciro
et al. (2014).

####  7.1.4 Observational Evidence for Radial Migration

Many papers have claimed observational evidence both for and against radial
migration but not all are equally convincing. So far, all evidence is
indirect, although one of the science goals of GALAH (GALactic Archeology with
HERMES; De Silva et al., 2015) and other upcoming Galactic spectroscopy
surveys is to use detailed chemical tagging to identify stars born from the
same molecular cloud, and to examine their distribution throughout the Milky
Way (but see also Casamiquela et al., 2021).

Three papers stand out: Kordopatis et al. (2015), using RAVE (Radial Velocity
Experiment) data (Steinmetz et al., 2006), found supersolar metallicity stars
having lowish eccentricity orbits in the solar neighborhood and argued they
must have migrated from the inner disk. Hayden et al. (2015), using Apache
Point Observatory Galactic Evolution Experiment (APOGEE) data (Majewski et
al., 2017), measured the metallicity distribution functions (MDFs) across a
large volume of the Milky Way disk having a radial and vertical extents of
$3<R<15\;$kpc and $|z|<2\;$kpc respectively. They found a striking systematic
change with radius to the shape of the midplane MDF and concluded that radial
migration was the most likely explanation for the shape of the MDF in the
outer Galaxy. Frankel et al. (2020) fitted a model of churning and blurring to
APOGEE red clump stars, concluding that the secular orbit evolution of the
disk is dominated by diffusion in angular momentum, with radial heating being
an order of magnitude lower.

###  7.2 Flattening Rotation Curves

The rotation curve, or circular speed as a function of radius, is remarkably
smooth for most galaxies (e.g. Lelli, McGaugh & Schombert, 2016, their Fig 5
in the html version only). There is barely a feature even where the central
attraction shifts from being baryon-dominated to dark matter-dominated, which
Kent (1987, p828) described as a “disk-halo conspiracy.” A few authors (e.g.
Kalnajs, 1983, Kent, 1986, Palunas & Williams, 2000) have drawn attention to
“bumps and wiggles” (Freeman, 1992, p205) in long-slit rotation curves, some
of which correspond to photometric features in the light profile. While this
is undeniable evidence for significant mass in the disk, the underlying cause
of these small-scale features may be spiral arm streaming rather than
substantial fluctuations in the radial mass profile of the disk.

Spiral instabilities may also be responsible for featureless rotation curves,
as first argued by Lovelace & Hohlfeld (1978). Berrier & Sellwood (2015)
conducted experiments with growing disks in which they artificially chose to
add material having a narrow range of angular momentum. Some of their models
had a dense central mass and all had a (rigid) cored outer halo. They found
that no matter what the angular momentum of the added particles, the mass
distribution in the disk rearranged itself such that the resulting rotation
curve became remarkably featureless.

####  7.2.1 Smoothing Mechanism

Berrier & Sellwood (2015) presented a more controlled experiment in which they
added particles to the stable Mestel disk to build a narrow ridge. The spirals
that developed in this model were the result of two unstable modes that were
provoked by the density ridge. Local stability analysis of an axisymmetric
ridge-like density excess (Sellwood & Kahn, 1991) predicts that, for each
sectoral harmonic, the normal modes are wave pairs with corotation on opposite
sides of the ridge. The most rapidly growing pair of modes was for $m=3$ in
their simulation, which was preferred by the disk supporting response (see
§4.2.3.2). As the mode amplitudes rose, horseshoe orbits (§7.1.1 and Figure
2a) developed at both CRs but, unlike in a featureless disk, the presence of
the ridge caused the resulting $L_{z}$ changes to be strongly out of balance
in opposite senses for each separate mode; naturally, the combined effect of
both modes did not change the total $L_{z}$ of the disk. Since CR scattering
removed far more particles from the ridge than were added to it, the ridge was
erased and the rotation curve was flattened almost perfectly.

####  7.2.2 Maximum Entropy State

Thus it seems that the distribution of angular momentum in the baryonic
material that created a galaxy disk does not need to be able to account for
the featureless character of most galaxy rotation curves. Any small-scale
variations in a reasonable distribution will be erased by spiral activity.

Herpich, Tremaine & Rix (2017) developed a maximum entropy formalism to
determine the expected surface density profile in a disk in which radial
migration efficiently scrambles the angular momenta of individual stars, while
preserving the circularity of their orbits and the total mass and angular
momentum of the disk. They showed that the maximum entropy surface mass
profile was not perfectly exponential, but nevertheless corresponded well with
the surface brightness profiles of a large sample of disk galaxies. Since disk
galaxies generally possess population and/or color gradients, they cannot have
fully reached the maximum-entropy end state, but nevertheless their mass
profiles appear to be close to this idealized model.

###  7.3 Driving Turbulence in the ISM

It has long been recognized (e.g. Rees, 1994), that the origin of the large-
scale magnetic field in galaxies presents a challenge, in that the standard
$(\alpha,\Omega)$ dynamo mechanism (Parker, 1955) has difficulties creating
large-scale fields of the observed strength from the expected seed fields. The
process uses differential rotation, the $\Omega$-effect, combined with
turbulence in the ISM, the $\alpha$-effect, to amplify the field. Calculations
(e.g. Ferrière, 1998) that invoke turbulence driven by mechanical input to the
ISM even from clustered supernovae struggle to achieve the observed field
strengths primarily because the turbulence is driven on too small a scale.
More recent work is thoroughly reviewed by Khoperskov & Khrapov (2018).

Figure 8: Part of the evolution of test particles that began on rings having
initially circular orbits in the groove-unstable model used by Sellwood &
Binney (2002). The rings of particles can be thought of as tracing streamlines
up until the moment they intersect. Reproduced from Sellwood & Preto (2002).

However, transient spiral waves churn not only the stellar distribution
(§7.1), but also the ISM. The three snapshots in Figure 8 (from Sellwood &
Preto, 2002) show part of the evolution of rings of test particles that began
on initially circular orbits in the groove-unstable model used by Sellwood &
Binney (2002). In the period shown, the growing instability causes distortions
to the rings that are most pronounced near the CR of the instability.
Imagining the non-interacting particle rings to trace gas streamlines, with
individual clouds threaded by magnetic field, it is clear that the evolving
spiral potential creates crossing streamlines, at which point collisions
between gas clouds would occur. Note that particles from widely differing
radii encounter each other, and that the spiral instability drives turbulence
on a much greater radial scale than would supernovae. Sellwood & Binney (2002)
therefore suggested that the slow magnetic field amplification from supernovae
alone could be accelerated by this source of turbulence on grander scales.

Unfortunately, this suggestion has yet to be subjected to a convincing test.
Although spiral-driven turbulence may well have contributed to the promising
magnetic field amplification reported by Pakmor et al. (2017), their
simulations included too many physical processes to be able to isolate the
role of non-axisymmetric gravitational forces arising from spiral arm
evolution. The simulations by Khoperskov & Khrapov (2018) included self-
gravity of the magnetized gas only, but adopted the gravitational field of an
imposed, steadily rotating spiral potential, which crucially omits the
evolving gravitational field that is important to driving turbulence by radial
migration, while Wibking & Krumholz (2021) employed a sub-maximum disk that
developed multi-arm spirals that are unable to drive large radial excursions
(§7.1.3).

###  7.4 Age-velocity Dispersion Relations

Wielen (1977), and others, pointed out long ago that the random motions of
older disk stars in the Milky Way are greater than those of younger ones. The
evidence was enormously strengthened and extended by Mackereth et al. (2019),
who made use of the stunning improvement to stellar kinematics from Gaia DR2,
abundance analysis from APOGEE, and state-of-the-art techniques to assign ages
to stars. Their sample, which they separate it into high and low $[\alpha/{\rm
Fe}]$, extends over a broad radius range and vertical distance from the mid-
plane. $[\alpha/{\rm Fe}]$A logarithmic measure, relative to solar values, of
the $\alpha$-element abundance relative to that of iron.

Figure 9: Dispersions of the vertical (a) and radial (b) stellar velocities
divided into age bins and color coded by Galacto-centric radius. The age bins
for each group of points are the same but the points have been shifted
slightly in mean age to separate them more clearly. (c) The ratio of vertical
to radial axes of the stellar velocity ellipsoid. Figure adapted from figs 5
and 7 of Mackereth et al. (2019), with permission.

Two of their principal findings are reproduced in Figure 9a,b clearly reveal
the kinematically distinct and older population of the high $[\alpha/{\rm
Fe}]$ stars, which in their sample are predominantly in the inner Milky Way.
Also, older stars have larger velocity spreads at all radii, with the high
$[\alpha/{\rm Fe}]$ stars having distinctly larger $\sigma_{R}$.101010The
third velocity dispersion component, $\sigma_{\phi}$, is strongly coupled to
$\sigma_{R}$ through the epicycle motions of stars (Binney & Tremaine, 2008)
and is therefore not independent. Mackereth et al. (2019) fitted power laws to
the variations of dispersion with age, finding an index of $\sim 0.3$ for
$\sigma_{R}$ that is almost independent of $[\alpha/{\rm Fe}]$. The bottom
panel reveals that the velocity ellipsoid of the low $[\alpha/{\rm Fe}]$ stars
is quite flattened in the inner disk, though less strongly with age, and
becomes much rounder at larger radii.

Sellwood (2014) reviewed the mechanisms that have been invoked to account for
the now firmly established rise in velocity dispersion with age, which include
spiral scattering, giant molecular cloud (GMC) scattering, SF in turbulent
gas, and the influence of tidal interactions. It is likely that all these
mechanisms, and perhaps others, contribute to the general increase of
dispersion with age, but he preferred the theory that it was largely driven by
spiral scattering, as was endorsed by Mackereth et al. (2019). Note that
resonant scattering by spirals can drive up the in-plane components of random
motion, but has little effect on the vertical component; therefore a
population of GMCs is needed to redirect the in-plane peculiar velocities into
the vertical direction.

The ratio of vertical to radial dispersions $\mathrel{\hbox to0.0pt{\lower
2.15277pt\hbox{$\mathchar 536\relax$}\hss}\raise 1.72218pt\hbox{$\mathchar
316\relax$}}0.5$ (Figure 9c) among the low $[\alpha/{\rm Fe}]$ stars at, and
interior to, the Solar circle is surprisingly low, since cloud scattering
should quickly lead to a value of $\sim 0.6$, as predicted by Ida et al.
(1993) for a flat rotation curve. Their prediction, which took proper account
of distant scattering and is in agreement with numerical results (Villumsen,
1985, Shiidsuka & Ida, 1999, Hänninen & Flynn, 2002, Sellwood, 2008), is that
the equilibrium axis ratio depends weakly on the local slope of the rotation
curve, with smaller values for declining, and larger for rising, $V_{c}$ with
radius.

Thus flatter values reported in Figure 9c could perhaps be an indicator of a
declining rotation curve, although Ida et al. (1993) expect a ratio as low as
$\sim 0.4$ only when the circular speed declines in a Keplerian fashion, which
seems unlikely. It is also possible that cloud scattering is inefficient such
that the expected equilibrium ratio has not been established, which would
require very few GMCs in the disk of the Milky Way. However, the ratios
reported by Mackereth et al. (2019) are lower than those found in previous
work (e.g. Holmberg et al., 2009) and, since their survey ranges over greater
distances from the Sun, could result from overestimates of $\sigma_{R}$ if the
second moment of the stellar velocity distribution includes spatial variations
of non-circular streaming motions due to the bar and spirals.

The rounder shape of the ellipsoid at larger radii, indicated by the red
points in Figure 9c, is inconsistent with any predicted slope of the rotation
curve, and is likely due to satellite heating, as also proposed by Mackereth
et al. (2019).

Finally, the dispersions of the older, high $[\alpha/{\rm Fe}]$ stars do not
appear to change with age, which could merely reflect uncertain age estimates.
Nevertheless, it is clear these stars constitute a kinematically distinct
population, and their random velocities are thought to have been created by
another mechanism, such as a minor merger as the Milky Way accreted a
satellite that thickened the then disk, or stars forming in turbulent gas in
the early stages of galaxy assembly (e.g. Genzel et al., 2008).

###  7.5 Galaxy Formation

Simulations of galaxy formation (e.g. Vogelsberger et al., 2020) are
proceeding apace and the resulting galaxy models have a more authentic
appearance with almost every new paper, although quantitative differences from
the properties of real galaxies remain. An attempt to survey this progress
would quickly become out-of-date, and would anyway be outside the realm of
this review. Here we confine ourselves to a few remarks related to spirals
within the thin disks of the model galaxies that constitute a challenge to the
simulators.

A thin disk component in the model galaxies of a few years ago could be
recognized only among the very young star particles and gas. A clear example
was given in Fig. 1 of Bird et al. (2013, p3), which separated a simulated
galaxy model at the present day into a number of stellar “age cohorts” and
presented face-on and edge-on projected densities of each cohort. Only the
youngest cohort, those stars that had formed within 0.5 Gyr of the moment of
analysis, were in a thin layer and manifested clear spiral patterns. The
surface density profiles and thicknesses of each separate cohort were
quantified in Fig. 2 of their paper, and the radial and vertical velocity
dispersions in their Fig. 4. The next youngest cohort having ages in the range
of 0.5 Gyr to 5.4 Gyr, had a much greater vertical thickness and weak spiral
patterns; no significant spirals could be discerned in the still older and
hotter cohorts. The surface density of the youngest cohort was just a few
percent of the total projected density and, consistent with swing-
amplification theory (§4.2.3.2), supported multi-armed spirals in a low-mass
cool disk. The weaker spirals in the second oldest, and more massive, cohort
had lower rotational symmetry, as theory would predict, but the greater
velocity dispersions and thickness inhibited strong patterns. Thus the low
mass and limited age range of the thin disk in their model was at variance
with what we know of the thin disk in the Milky Way (Bland-Hawthorn & Gerhard,
2016), and nature of the spirals in their model bear little resemblance to
those in most galaxies (§2). The fraction of the disk mass in a thin component
in the very recent FIRE-2 (Feedback in Realistic Environments) models (Yu et
al., 2021) reached $\sim 50$% in a few cases, but this is still not as large
as it should be.

The consequences of disks that are too hot and thick in the simulated galaxies
are clear. Spirals patterns tend to be unrealistically weak and/or multi-
armed, which has two consequences: (1) radial migration is reduced, as
reported by Bird et al. (2013) and by Avila-Reese et al. (2018), for the
reasons given in §7.1.3. (2) Lindblad resonance scattering plays a lesser role
than it should in disk heating (eq. 6), although other mechanisms have clearly
created rather too much random motion. Furthermore, the peculiar velocities of
the disk stars in real galaxies are re-oriented by scattering off molecular
clouds, whereas in galaxy formation simulations collisional relaxation due to
supermassive particles (e.g. Sellwood, 2014, Ludlow et al., 2021) will have
the same effect for the wrong reason!

##  8 SUMMARY AND CONCLUSIONS

The ubiquity of spiral patterns in the stellar disks of galaxies requires them
to result from self-excited instabilities within the disk. Other mechanisms,
such as bars and tidal encounters, may well drive spiral responses in specific
cases, but we concluded in §3 that such external driving could not account for
all, or perhaps even most, spiral patterns.

The self-excitation mechanism in simulations of isolated, unbarred disk galaxy
models is now established to be a recurrent cycle of groove modes (§5.1).
Individual modes, which have constant pattern speed at all radii, grow and
decay, with each having significant amplitude for just a few turns at its
corotation radius, while new instabilities develop to maintain spiral
activity. However, the superposition of several co-existing modes causes the
spiral appearance to change rapidly and the arms to appear to wind up over
time. We argue that other theories have weaknesses (§5.2), and propose that a
groove-mode cycle could be responsible for spirals in real galaxies, as well
as in simulations. Observations (§2) suggest that swing amplification (§4.2.3)
plays a role in spiral formation, which is a mechanism at the root of most
theories, while evidence specific to a recurrent cycle of groove modes is hard
to obtain. We have only hints from the Gaia DR2 data that the distribution of
disk stars in the Milky Way (§5.3) manifests some of the features expected
from a groove-mode cycle.

The recurring patterns cause a secular increase in the random motions of stars
in the disk, reducing its responsiveness to subsequent instabilities, and
spiral activity in a purely stellar disk must fade over time. However, spiral
activity can be maintained indefinitely if the disk has even a modest fraction
of gas, since gas clouds are able to maintain a low velocity dispersion
through dissipative collisions, and form stars sharing similar kinematic
properties, thereby maintaining the responsiveness of the disk (§6.1).

Spiral activity is a major driver of secular evolution in disk galaxies. It
churns the disk stars, causing a radial diffusion that flattens metallicity
gradients (§7.1). It also erases density features in the disk (§7.2), implying
that the smoothness of density profiles and of rotation curves need not be
properties that are required of galaxy formation. The scattering of stars at
Lindblad resonances causes a secular rise in the in-plane components of random
motions, which can be scattered by GMCs into the vertical direction. These
processes must contribute to the observed increase in random motions of disk
stars with age within the Milky Way (§7.4) and in other galaxies.

Simulations of sub-maximum disks do not manifest spiral patterns that are
typical of most galaxies (§2.2), and their multi-arm features do not capture
the full spiral-driven evolution of disk galaxies (§7.1.3). These shortcomings
are shared by galaxy formation simulations that have not yet succeeded in
creating thin disks that are cool and massive enough to support realistic
spiral patterns (§7.5).

While we have reviewed the steady progress that has been made in the
development of our understanding of disk galaxy dynamics, we look forward to
more and better observational data to test these ideas (§2.8). Also, a number
of outstanding theoretical issues remain to be settled, which include the
following:

  1. 1.

Foremost is the problem of the stability of disks having gently rising
rotation curves (§4.4.1.2). Despite years of effort, we have no satisfactory
explanation for the absence of bars in such galaxies that often seem to
manifest two-arm spiral patterns.

  2. 2.

We have only a few specific models of spirals being tidally driven, and we
need to know the mass range and orbit parameters of encounters with companion
galaxies that can excite a spiral response without triggering a bar (§3).

  3. 3.

We still lack compelling evidence that the recurrent cycle of groove modes,
which has been identified as the mechanism for spiral generation in
simulations (§5.1), also works in galaxies. The later releases of Gaia data
may yield stronger evidence in the Milky Way, but additional evidence from
external galaxies would be highly desirable.

  4. 4.

Transient spiral instabilities in heavy disks drive large-scale turbulence in
the gas component in galaxies, which should strongly enhance magnetic field
growth. However, no direct tests of this prediction have yet been made (§7.3).

  5. 5.

Our discussion of the origin and effects of spiral patterns has concentrated
on isolated galaxies, which present the most compelling need for a theoretical
explanation. We find on-going cosmological gas infall is needed to maintain
self-excited spiral activity (§6.1) and recognize that hierarchical clustering
drives some spirals by tidal encounters (§3). But the numerically challenging
simulations of fully cosmological galaxy formation have not yet created cool
stellar disks massive enough to support bi-symmetric spirals that are common
in the nearby universe (§7.5). Once that is achieved, we will be able to test
whether the spiral dynamics discussed in this review applies in the full
cosmological context.

## DISCLOSURE STATEMENT

The authors are not aware of any affiliations, memberships, funding, or
financial holdings that might be perceived as affecting the objectivity of
this review.

## ACKNOWLEDGMENTS

We thank Scott Tremaine, John Kormendy, and the editor, Rob Kennicutt, for
many highly valuable comments on earlier drafts of this review. Other comments
from Ray Carlberg, Agris Kalnajs, and Igor Pasha were also helpful. JAS
acknowledges the continuing hospitality of Steward Observatory.

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