# Prograde and meandering wall modes in rotating Rayleigh-Bénard convection
with conducting walls

S. Ravichandran  Nordita, KTH Royal Institute of Technology and Stockholm
University, Stockholm 10691, Sweden  Interdisciplinary Programme in Climate
Studies, Indian Institute of Technology Bombay, Mumbai 400076, India  J. S.
Wettlaufer  Nordita, KTH Royal Institute of Technology and Stockholm
University, Stockholm 10691, Sweden  Yale University, New Haven, Connecticut
06520-8109, USA

###### Abstract

We study numerically the onset of convection in rotating Rayleigh-Bénard
convection in horizontally confined geometries of aspect ratio $2$ with the
walls held at fixed temperatures. We show that this arrangement is
unconditionally unstable to flow that takes the form of wall-adjacent
convection rolls. For wall temperatures close to the temperatures of the upper
or lower boundaries, we show that the base state undergoes a Hopf bifurcation
to a state comprised of spatio-temporal oscillations–‘wall modes’–precessing
in a retrograde direction. We show that the velocity boundary conditions at
the upper and lower boundaries are crucial to the formation and propagation of
the wall modes: asymmetric velocity boundary conditions at the upper and lower
boundaries can lead to prograde wall modes, while stress-free boundary
conditions at both walls can lead to wall modes that have no preferred
direction of propagation.

##  I Introduction

Rotating Rayleigh-Bénard convection provides a model framework to study
processes in stars (Schumacher and Sreenivasan, 2020), protoplanetary disks
(Gerbig and Laughlin, 2022), planetary cores and mantles (Aurnou  _et al._ ,
2015; Bercovici, 2015; Aurnou  _et al._ , 2020), and the cryosphere (e.g.
Wells and Wettlaufer, 2008; Ravichandran and Wettlaufer, 2021). Moreover,
rotating Rayleigh-Bénard convection acts as a model system for the study of
pattern formation, such as transient axisymmetric rings observed during spin-
up (e.g., Boubnov and Golitsyn, 1986; Vorobieff and Ecke, 1998; Zhong  _et
al._ , 2010; Ravichandran and Wettlaufer, 2020), and travelling waves observed
near the onset of convection (Ecke  _et al._ , 1992; Ning and Ecke, 1993; Liu
and Ecke, 1997, 1999). In typical model studies of rotating Rayleigh-Bénard
convection, both theory and numerical simulations employ horizontally periodic
domains, whereas experiments do not, in part leading to disagreements. For
instance, Rossby (1969) observed that in rotating Rayleigh-Bénard convection
with the typical no-slip experimental boundary conditions, the onset of
convection occurs for significantly smaller thermal forcing than predicted by
the linear stability analysis of Chandrasekhar (1953). Although explanations
of this apparent disagreement between theory and experiment can be traced to
the basic differences between boundary conditions and finite amplitude
perturbations (e.g., Veronis, 1966, 1968; Herrmann and Busse, 1993; Kuo and
Cross, 1993), there are still many outstanding questions (e.g., Ecke, 2023).
For example, measurements suggest that the peripheral modes may also be
responsible for the mismatch between the Nusselt numbers in laboratory
experiments and numerical simulations (Kunnen  _et al._ , 2013; Ecke  _et al._
, 2021).

In the absence of rotation, the onset of convection in a horizontally
unbounded layer of fluid of depth $H$ across which a temperature difference
$\Delta T$ is maintained, is governed by the dimensionless Rayleigh number

$Ra=\frac{g\alpha\Delta TH^{3}}{\nu\kappa},$ (1)

where $g$ is the acceleration of gravity and $\nu$ and $\kappa$ are the
viscosity and thermal diffusivity of the fluid. The onset of convection occurs
when a critical Rayleigh number, $Ra_{c}^{\mathrm{\tiny{bulk}}}$, is exceeded,
where $Ra_{c}^{\mathrm{\tiny{bulk}}}$ = $O(10^{3})$, with the exact value
depending on the boundary conditions. Chandrasekhar (1953) showed that,
independent of the Prandtl number $Pr=\nu/\kappa$, the instability that leads
to convection is non-oscillatory.

Rotation about the vertical axis suppresses the effects of buoyancy and thus
enhances stability. Therefore, the critical Rayleigh number increases with the
rotation rate, $\Omega$, as

$Ra_{c}^{\mathrm{\tiny{bulk}}}\sim E^{-4/3},$ (2)

where $E=\nu/2\Omega H^{2}$ is the Ekman number (Chandrasekhar, 1953; Veronis,
1966). In contrast to non-rotating systems, the onset of convection in
horizontally unbounded rotating Rayleigh-Bénard convection can be oscillatory
if the Prandtl number $Pr<0.69$ (Chandrasekhar, 1953).

Rossby (1969) showed experimentally that convection sets in for much smaller
Rayleigh numbers than predicted by Eq. (2), the possible origins of which were
discussed contemporaneously by Veronis (1968) as being associated with
boundary conditions. Of relevance to our study, the experiments by Ecke  _et
al._ (1992), supported by linear stability analysis by Goldstein  _et al._
(1993), showed that the wall-adjacent convection takes the form of a
travelling wave with a phase speed opposite to the sense of rotation. They
also showed that the system undergoes a Hopf bifurcation at a critical
Rayleigh number, following which the travelling wave appears. These results
have been extended to Rayleigh numbers $O\left(10^{5}\right)$ by Lopez  _et
al._ (2007).

Herrmann and Busse (1993) and Kuo and Cross (1993) independently showed that,
in the asymptotic limit of $E\rightarrow 0$, the critical Rayleigh number for
the onset of the travelling waves in confined rotating Rayleigh-Bénard
convection with adiabatic walls is

$Ra_{cw}^{\mathrm{\tiny{ZF}}}=\pi^{2}(6\sqrt{3})^{1/2}E^{-1}=31.82E^{-1},$ (3)

where the superscript ZF denotes ‘zero flux’. Herrmann and Busse (1993) also
showed that conducting walls stabilise the wall modes, with a critical
Rayleigh number that has the same leading order scaling as that for an
infinite layer, but with a smaller prefactor:

$Ra_{cw}^{\mathrm{\tiny{FT}}}=0.9086\left(\pi
E^{-1}\right)^{4/3}+2.124\left(\pi
E^{-1}\right)^{7/6}=4.18E^{-4/3}+8.08E^{-7/6},$ (4)

where the superscript FT denotes ‘fixed temperature’ highlighting the fact
that mathematically well-posedness requires the temperature to be prescribed
at a conducting boundary. In their analysis, Herrmann and Busse (1993) assume
a wall temperature equal to the purely conductive (linear) profile

$\bar{T}_{\mathrm{\tiny{wall}}}=1-z.$ (5)

The flow, for $Ra>Ra_{cw}^{\mathrm{\tiny{FT}}}$, again takes the form of
travelling waves that propagate in a retrograde sense, against the sense of
rotation.

If, instead of the linear temperature boundary condition (BC) of Herrmann and
Busse (1993), a constant temperature is imposed across the entire height of
the walls, the system is unconditionally unstable with the flow taking the
form of wall-adjacent convective rolls. In this study, we examine the effects
of changing the uniform temperature at which the walls are held, varying the
temperature between those of the upper and lower boundaries. We show that the
steady roll state undergoes a Hopf bifurcation as a funtion of the wall
temperature, leading to wall modes. We also show that the velocity BCs at the
upper and lower boundaries control the direction of propagation of the wall
modes, and prograde wall modes can arise for suitable velocity BCs.

The rest of the paper is organised as follows. In §II, we set up the geometry
of the problem and describe the numerical method used for the simulations. In
§III, we present results from the numerical simulations as the wall
temperature and the velocity BC at the upper boundary are varied, compare our
results with other known travelling wave solutions in rotating Rayleigh-Bénard
convection, and discuss the effects of asymmetric velocity BCs at the upper
and lower boundaries on the direction of propagation of the wall-adjacent
spatio-temporal patterns. We conclude in §IV.

##  II Setup and Numerical Simulations

The domain is a cubic volume of height $H$ and width $L$, with an aspect ratio
$\Gamma=L/H=2$, shown schematically in figure 1. The system rotates about the
vertical $z-$axis with a constant angular velocity $\Omega$.

Figure 1: Our simulations are performed in a cuboidal volume of square cross-
section and aspect ratio $L/H=2$. The walls are either adiabatic or held at a
constant temperature. The container rotates about the vertical axis with a
constant angular velocity, with gravity pointing vertically down.

We make the Boussinesq approximation, so that fluid properties are assumed to
be constant and the flow is assumed to be incompressible. We nondimensionalise
the governing equations using the length scale $H$ and the buoyancy velocity
scale $U_{b}=(g\alpha\Delta TH)^{1/2}$. The nondimensional governing equations
become

$\displaystyle\frac{D\boldsymbol{u}}{Dt}$ $\displaystyle=-\nabla
p-2Ro^{-1}\boldsymbol{e}_{z}\times\boldsymbol{u}+\left(\frac{Pr}{Ra}\right)^{1/2}\nabla^{2}\boldsymbol{u}+\boldsymbol{e}_{z}\theta,$
(6) $\displaystyle\nabla\cdot\boldsymbol{u}$ $\displaystyle=0,$ (7)
$\displaystyle\frac{D\theta}{Dt}$
$\displaystyle=\left(\frac{1}{RaPr}\right)^{1/2}\nabla^{2}\theta,$ (8)

where $Ro=2E\left(Ra/Pr\right)^{1/2}$ is the Rossby number.

We consider the combinations of BCs listed in table 1.

BC | $z=0$ | $z=1$ | Walls
---|---|---|---
SYMNS |  $u=v=w=0$, $\theta=1$ |  $u=v=w=0$, $\theta=0$ |  $u=v=w=0$, $\theta=\theta_{w}$ or $\partial\theta/\partial n=0$
ASYM |  $u=v=w=0$, $\theta=1$ | $w=0,\partial(u,v)/\partial z=0$ |  $u=v=w=0$, $\theta=\theta_{w}$ or $\partial\theta/\partial n=0$
Table 1:  The combinations of boundary conditions (BCs) considered. Symmetric
(‘SYMNS’) BCs have no-slip upper and lower boundaries, while asymmetric
(‘ASYM’) BCs have free-slip upper and no-slip lower boundaries. Consequences
of symmetric free-slip (‘SYMFS’) BCs, defined in analogy with SYMNS BCs, and
of free-slip BCs at all boundaries (‘ALLFS’) are examined in §III.2.

The temperature of the lower boundary is fixed at $\theta(z=0)=1$, the upper
boundary at $\theta(z=1)=0$, and the walls are either adiabatic
($\partial\theta/\partial n=0$) or have a fixed temperature
$\theta=\theta_{w}$, with $0\leq\theta_{w}\leq 1$. The velocity obeys either
no-slip or stress-free BCs at the walls at $x=\pm 1,y=\pm 1$, and the lower
and upper boundaries at $z=0$ and $z=1$ respectively.

Subject to these BCs, equations (6–8) are solved using the finite volume
solver _Megha_ -5, used in previous studies of convection (Ravichandran and
Wettlaufer, 2020, 2021; Ravichandran  _et al._ , 2022). The solver uses
second-order central differences in space and a second-order Adams-Bashforth
timestepping scheme. The results presented are grid-independent, with a grid
resolution of up to $256^{3}$ uniformly spaced points in the three space
directions and a timestep $dt\geq 1.25\times 10^{-3}$.

##  III Results and Discussion

Equations (6–8) have as governing parameters the nondimensional numbers $E$,
$Ra$ and $Pr$. Additionally, $\theta_{w}$ and the aspect ratio $\Gamma$ enter
as parameters in the boundary conditions. We minimise the amount of
computation required by exploring the effects of varying the parameters one at
a time around the point ($E=10^{-4}$, $Ra=10^{6}$, $Pr=1$, $\Gamma=2$) in
parameter space, with $0\leq\theta_{w}\leq 1$. For this combination of $E$ and
$Pr$ in a horizontally unbounded geometry, we have
$Ra_{c}^{\mathrm{\tiny{bulk}}}\approx 1.5\times 10^{6}$, while with insulating
walls, wall modes appear for $Ra>Ra_{cw}^{\mathrm{\tiny{ZF}}}=3.2\times
10^{5}$. For walls with the linear temperature profile (equation 5),
$Ra_{cw}^{\mathrm{\tiny{FT}}}=1.3\times 10^{6}$.

###  III.1 Symmetric and asymmetric velocity boundary conditions

We first examine the flow structures that arise with SYMNS BCs for
$E=10^{-4},Ra=10^{6},Pr=1,\Gamma=2$, while varying $\theta_{w}$. Two types of
flow may arise, as shown in figure 2; a wall temperature $\theta_{w}=0.25$
leads to steady wall-attached rolls in the shape of the container, while
$\theta_{w}=0.1$ generates retrograde wall modes.

Figure 2:  The steady convective state with wall-adjacent rolls for
$E=10^{-4}$, $Pr=1$, $Ra=10^{6}$ and SYMNS BCs for (a,c) $\theta_{w}=0.25$ and
(b,d) $\theta_{w}=0.1$. The subplots show the horizontal (a,b) and vertical
(c,d) cross-sections of the vertical velocity $w$. In (a,c), a steady state of
rolls is reached.

The retrograde propagation of the wall modes is apparent from the Hövmöller
diagram in figure 3(a), where we show the near-wall temperature along a
horizontal line. Due to the symmetry of the problem, retrograde wall modes
with a similar structure are observed for $1-\theta_{w}=0.1$, as seen in
figure 3(b).

Figure 3:  Hövmöller plots of the temperature $\theta$ for
$E=10^{-4},Ra=10^{6},Pr=1$, and (a) $\theta_{w}=0.1$ and (b) $\theta_{w}=0.9$.
The slope of the patterns shows that the wall modes propagate in a retrograde
direction. The symmetry of the Boussinesq equations ensures that the flow
patterns for $\theta_{w}$ and $1-\theta_{w}$ are similar.

Wall modes are also seen with ASYM BCs for wall temperatures close to upper or
lower boundary values, as seen in figures 4. However, the behaviours for
$\theta_{w}\rightarrow 0$ and $\theta_{w}\rightarrow 1$ are not symmetrical.
Instead, we find that while retrograde wall modes exist for
$\theta_{w}\rightarrow 0$ (figure 5(a)), the wall modes travel in a prograde
direction for $\theta_{w}\rightarrow 1$ (figure 5(b)). Convection in steady
rolls is seen for intermediate $\theta_{w}$, as with SYMNS BCs.

Figure 4:  The wall modes observed for $E=10^{-4}$, $Pr=1$, $Ra=10^{6}$ and
ASYM BCs for (a,c) $\theta_{w}=0.05$ and (b,d) $\theta_{w}=1$. The subplots
show the horizontal (a,c) and vertical (b,d) cross-sections of the vertical
velocity $w$. In (a,c), the wall modes propagate in the retrograde direction,
while in (b,d) they propagate in the prograde direction (see figure 5).

In figure 5, we plot the Hövmöller diagrams for the space-time evolution of
the wall modes, showing this reversal of the direction of propagation.

Figure 5:  Hövmöller plots of the temperature $\theta$ for the same parameters
as in figures 4, showing that (a) for $\theta_{w}=0.05$ the wall modes travel
in the retrograde direction, while (b) for $\theta_{w}=1$ they travel in the
prograde direction.

Horn and Schmid (2017) find flow features that propagate in the prograde
direction for small aspect ratios ($\Gamma=0.5<1$). These flow features,
however, are ‘bulk modes’, and not restricted to the near-wall region. These
modes are related to the slow modes of Goldstein  _et al._ (1993), with
precession frequencies much smaller than those of the retrograde modes.
Neither of these studies considers the effect of asymmetric velocity BCs,
although Goldstein  _et al._ (1993) do consider conducting walls. Here, we see
prograde _wall modes_ with asymmetric velocity BCs, but only when
$\theta_{w}\rightarrow 1$. In contrast, for insulating walls, the wall modes
are retrograde even for asymmetric BCs; and if the wall temperature is very
different from that of the no-slip boundary ($\theta_{w}\lesssim 0.9$, say),
the flow is comprised of nested rolls.

This behaviour can be explained by examining the vertical velocity at
conducting (figure 4) and insulating (not shown) walls. At conducting walls
with $\theta_{w}\rightarrow 1$, the near-wall flow has a positive vertical
velocity. When a sufficiently strong upward-moving flow reaches the upper
boundary, it turns towards the axis of rotation by continuity. In the rotating
frame of reference, the Coriolis effect tends to force the flow to the right,
in the prograde direction. Since the buoyancy is proportional to the
temperature, a sufficiently large upward velocity only results when
$\theta_{w}\rightarrow 1$. If the walls are insulating, this mechanism is no
longer active, owing to the absence of a net upward flow in the near-wall
region.

###  III.2 Additional combinations of boundary conditions

We consider two other combinations of BCs, briefly alluded to in §II. If both
the upper and lower boundaries are stress-free, and the walls are conducting
and obey the no-slip condition (‘SYMFS’ BCs, see table 1), wall modes form but
have no clearly discernible direction of propagation. Insulating walls lead to
the standard retrograde wall modes, in agreement with earlier studies.
Similarly, if both the upper and lower boundaries as well as the conducting
walls are stress-free (‘ALLFS’ BCs), the wall modes that appear have no fixed
direction of propagation. Representative snapshots of the wall modes that
result from SYMFS and ALLFS BCs are shown in figures 6(a) and (b)
respectively, with the corresponding Hövmöller diagrams plotted in figures
7(a) and (b) respectively.

Figure 6:  The wall modes observed for $E=10^{-4}$, $Ra=10^{6}$, $Pr=1$ and
(a,c) SYMFS BCs with $\theta_{w}=0.95$ and (b,d) ALLFS BCs with
$\theta_{w}=0.1$. The subplots show the horizontal (a,b) and vertical (c,d)
cross-sections of the vertical velocity $w$. The wall modes have no fixed
direction of propagation (see also figure 7).  Figure 7:  Hövmöller plots of
the temperature $\theta$ for the same parameters as in figures 6, showing that
for both (a) $\theta_{w}=0.95$ and SYMFS BCs, and (b) $\theta_{w}=0.1$ and
ALLFS BCs, the wall modes show no fixed direction of propagation.

A comparison of these figures to the equivalent figures for SYMNS (figures 2
and 3) and ASYM BCs (figures 4 and 5) suggests that while wall modes are
observed with conducting walls for suitable wall temperatures $\theta_{w}$,
no-slip velocity BCs on at least the upper or lower boundaries are necessary
for wall modes to propagate in a definite direction. This behaviour can again
be explained by flow continuity at the edges of the cubic domain between the
walls and the upper and lower boundaries where, due to the Coriolis effect,
the flow turns in opposite directions. In the geostrophic regime, flow
structures are also vertically homogeneous, and these tendencies cancel so
that the near-wall flow patterns have no fixed propagation direction.

###  III.3 Limits on $\theta_{w}$ for wall mode formation

From the results presented in §III.1 and §III.2, we see that wall modes only
occur for values of $\theta_{w}$ sufficiently close to the upper or lower
boundary temperatures, with the flow taking the form of steady rolls (see
figure 2(a,c)) for $\theta_{w}>\theta_{c}^{(1)}$. To determine the wall
temperature $\theta_{c}^{(1)}$ such that wall modes only exist for
$\theta_{w}<\theta_{c}^{(1)}$, we quantify the strength of the wall modes
using the oscillation amplitude of the temperature $\theta(x,y,z,t)$ as

$A_{\theta}^{2}=\langle\theta-\bar{\theta}\rangle,$ (9)

where $\bar{\theta}\left(x,y,z\right)$ is the time-averaged temperature at a
given location $(x,y,z)$, and the angular brackets $\langle\cdot\rangle$
denote an average over the spatial domain. Thus, the amplitude $A_{\theta}=0$
for steady (or zero) flow, and the wall temperature $\theta_{w}$ at which
$A_{\theta}>0$ is the critical wall temperature $\theta_{c}^{(1)}$. Similarly,
we define the temperature $\theta_{c}^{(2)}$ such that wall modes occur for
$\theta_{w}>\theta_{c}^{(2)}$, with steady convective rolls for
$\theta_{w}<\theta_{c}^{(2)}$.

In figure 8, we plot the oscillation amplitude $A_{\theta}$ as the wall
temperature $\theta_{w}$ is varied for SYMNS BCs and fixed flow parameters
$E=10^{-4},Ra=10^{6}$, and $Pr=1$. We find that the amplitude
$A_{\theta}^{2}\sim|\theta_{w}-\theta_{c}^{(1)}|$, with a critical wall
temperature $\theta_{c}^{(1)}\approx 0.1$. This dependence of the amplitude on
the supercriticality is similar to the Hopf bifurcation that occurs for wall
modes with insulating walls (Ecke  _et al._ , 1992), in which the controlling
parameter is the Rayleigh number.

Figure 8:  (a) The amplitude $A_{\theta}^{2}$ of the wall modes as a function
of time, showing the existence of a steady oscillatory state, and (b) the
domain-averaged oscillation amplitude $\bar{A}_{\theta}^{2}$ as a function of
the wall temperature $\theta_{w}$ for SYMNS BCs (see table 1). The curves in
(a) are plotted for the wall temperatures
$\theta_{w}=(0,0.05,0.1,0.1125,0.125,0.15,0.25)$, with circles of larger size
for smaller $\theta_{w}$. The averages in (b) are obtained over the last $100$
flow units of each curve in (a). The linear dependence of the (squared)
amplitude on the supercriticality is characteristic of a Hopf bifurcation.

Now we repeat this exercise for different values of $Ra$, keeping the
parameters $E=10^{-4}$ and $Pr=1$ fixed. The smallest value of $Ra$ such that
$\theta_{c}^{(1)}(Ra)>0$ is defined to be $Ra_{c}^{(1)}(E,Pr,\mathrm{BCs})$.
The resulting behaviour is summarised schematically in figure 9 where, since
we have only simulated a limited number of cases, we show a linear dependence
$\theta_{c}^{(1)}(Ra)$.

Figure 9:  The different regimes of behaviour observed with conducting walls
and SYMNS BCs. For small Rayleigh numbers $Ra$, wall-attached steady rolls are
seen for all $\theta_{w}$. For Rayleigh numbers greater than a critical value
$Ra_{c}^{(1)}\left(E,Pr,\Gamma\right)$, wall modes are seen for
$\theta_{w}<\theta_{c}^{(1)}$ and $\theta_{w}>\theta_{c}^{(2)}$, while the
steady roll state persists for $\theta_{c}^{(1)}<\theta_{w}<\theta_{c}^{(2)}$.
For larger Rayleigh numbers, convection also sets in away from the walls, with
the flow taking the form of a series of nested rolls that remain stable for
extended periods of time. For sufficiently large
$Ra>Ra_{c}^{(2)}=Ra_{c}^{\mathrm{\tiny{bulk}}}$, the flow takes the form of
horizontally drifting columnar vortices typical of rotating Rayleigh-Bénard
convection. For $E=10^{-4},Pr=1$, we find $Ra_{c}^{(1)}\approx 9\times
10^{5}$, and $Ra_{c}^{\mathrm{\tiny{bulk}}}=1.5\times 10^{6}$.

For SYMNS BCs and wall temperatures $\theta_{w}<\theta_{c}^{(1)}$, retrograde
wall modes are observed, as seen in figure 3(a). The symmetry of the problem
requires that $\theta_{c}^{(1)}+\theta_{c}^{(2)}=1$, with retrograde wall
modes for $\theta_{w}>\theta_{c}^{(2)}$ (See e.g., figure 3(b)). The
difference $\theta_{c}^{(2)}-\theta_{c}^{(1)}$ decreases continuously with
increasing $Ra$.

For Rayleigh numbers approaching the critical Rayleigh number
$Ra_{c}^{\mathrm{\tiny{bulk}}}\approx 1.5\times 10^{6}$ for the onset of flow
in the bulk, the difference $\theta_{c}^{(2)}-\theta_{c}^{(1)}$ can either (a)
decrease to zero before bulk flow sets in, such that
$Ra_{c}^{(2)}<Ra_{c}^{\mathrm{\tiny{bulk}}}$, or (b) remain finite until bulk
convection sets in at $Ra_{c}^{(2)}=Ra_{c}^{\mathrm{\tiny{bulk}}}$. We find
the latter case; specifically, we observe that with increasing $Ra$, the flow
takes the form of a series of nested convection rolls that span the entire
horizontal area of the domain, as shown in figures 10(a,c).

Figure 10:  With $E=10^{-4}$, $Pr=1$, and $\theta_{w}=0.4$, (a,c) steady
nested convection rolls that span the horizontal extent of the domain, for
$Ra=1.5\times 10^{6}$; and (b,d) the breakdown of the convection rolls into
individual vortices and bulk convection for $Ra=1.6\times 10^{6}$.

For $Ra>Ra_{c}^{\mathrm{\tiny{bulk}}}$, these rolls break down into individual
columnar vortices in the fluid bulk, as seen in figure 10(b,d) due to the
mechanism reported by Boubnov and Golitsyn (1986); Zhong  _et al._ (2010) and
Ravichandran and Wettlaufer (2020), and a state of geostrophic convection
exists in the bulk. For larger $Ra$, the wall modes become less prominent.

The symmetry between $\theta_{c}^{(1)}$ and $\theta_{c}^{(2)}$ observed for
SYMNS BCs breaks down if the velocity boundary conditions are asymmetric, e.g.
for ASYM BCs. In figure 11, we plot the oscillation amplitude
$A_{\theta}^{2}(\theta_{w})$ for the same parameters as in figure 8, but with
ASYM BCs, showing that $1-\theta_{c}^{(2)}<\theta_{c}^{(1)}$.

Figure 11:  (a) The amplitude $A_{\theta}^{2}$ of the spatio-temporal
oscillations as a function of time, showing the existence of a steady
oscillatory state, and (b) the domain-averaged oscillation amplitude
$\bar{A}_{\theta}^{2}$ as a function of the wall temperature $\theta_{w}$ for
ASYM BCs (see table 1). The curves in (a) are plotted for the wall
temperatures  $\theta_{w}=(0,0.05,0.1,0.15)$ (solid lines, circles of
decreasing size) and $\theta_{w}=(0.85,0.9,0.95,0.975,1.0)$ (dashed lines,
squares of increasing size). The averages in (b) are obtained over the last
100 flow units for each curve in (a). The linear dependence of the (squared)
amplitude on the supercriticality is characteristic of a Hopf bifurcation.
Note the asymmetry between $\theta_{c}^{(1)}$ and $\theta_{c}^{(2)}$ (c.f.
figure 8).

In the limit that $E\rightarrow 0$, equations (3) and (4) for the critical
Rayleigh numbers are independent of the Prandtl number. For finite $E>0$,
Herrmann and Busse (1993) found that smaller Prandtl numbers lead to smaller
$Ra_{c}$ (see their figures 3 and 6). Therefore, increasing the Prandtl number
has the same effect as increasing the Rayleigh number, and the system moves
closer towards instability; in other words, all other things being equal, when
increasing $Pr$, $Ra_{c}^{(1)}$ decreases and $\theta_{c}^{(1)}$ increases.
Similarly, decreasing the Ekman number $E$ increases the critical Rayleigh
number, and leads to the same qualitative effects as does decreasing $Ra$.

##  IV Conclusion

The most unstable modes in confined rotating Rayleigh-Bénard convection are
comprised of alternating regions of upwelling warm and downwelling cold fluid.
These patterns, commonly referred to as ‘wall-modes’, were first observed in
laboratory experiments. Here, in aspect ratios greater than unity, we have
used direct numerical simulations to study the formation and spatio-temporal
evolution of wall-adjacent flow patterns when the walls are conducting instead
of insulating. We showed that the wall temperature $\theta_{w}$ controls
whether the flow takes the form of steady convective rolls or propagating
wall-modes. We found that the velocity BCs are crucial to the dynamics of wall
modes, and that these modes propagate in a fixed direction only only if at
least one of the upper and lower boundaries obeys the no-slip condition.
Moreover, the direction of propagation of the wall modes can be reversed for
suitable combinations of the velocity BCs at the upper and lower boundaries
and the wall temperature. Particularly, if only the upper boundary is stress-
free, the wall modes propagate in a retrograde direction when
$\theta_{w}\approx 1$. Indeed, there is a similarity between the effects of
velocity BCs on wall mode formation and propagation, to transient convective
ring formation (Boubnov and Golitsyn, 1986; Vorobieff and Ecke, 1998; Zhong
_et al._ , 2010; Ravichandran and Wettlaufer, 2020).

Although wall modes were previously shown to occur in the presence of
conducting walls, we found that asymmetric velocity BCs drive an asymmetry in
the torques exerted at the two boundaries. Such effects may have astro- or
geophysical consequences, where asymmetric velocity BCs are common (e.g.
bodies of water with free upper surfaces). The effects of the asymmetry in
velocity BCs on integral flow properties such as the helicity are a subject of
ongoing study.

Finally, the rich range of flow behaviour in slightly supercritical Rayleigh-
Bénard convection has served as a model for the study of nonlinear-dynamical
systems, such as the Benjamin-Feir and Eckhaus instabilities (Ning and Ecke,
1993; Liu and Ecke, 1997, 1999; Lopez  _et al._ , 2009). More recent studies
have shown that the wall-modes in rotating Rayleigh-Bénard convection are a
topologically conserved feature, robust to severe modifications to the
geometry (Favier and Knobloch, 2020). Therefore, the influence of conducting
walls in combination with asymmetric velocity BCs on the wall modes may
provide a framework of general interest in the theory of pattern formation.

## Acknowledgements

Computational resources from the Swedish National Infrastructure for Computing
(SNIC) under grants SNIC/2020-5-471 and SNIC/2021-5-449 are gratefully
acknowledged. Computations were performed on Tetralith. The Swedish Research
Council, under grant no. 638-2013-9243, is gratefully acknowledged for
support. Nordita is partially supported by Nordforsk.

## Declaration of interests.

The authors report no conflict of interest.

## References

  * Schumacher and Sreenivasan (2020) J. Schumacher and K. R. Sreenivasan, Rev. Mod. Phys. 92, 041001 (2020).
  * Gerbig and Laughlin (2022) K. Gerbig and G. Laughlin, Astrophys. J. 930, 68 (2022).
  * Aurnou  _et al._ (2015) J. M. Aurnou, M. A. Calkins, J. S. Cheng, K. Julien, E. M. King, D. Nieves, K. M. Soderlund, and S. Stellmach, Phys. Earth Planet. Inter. 246, 52 (2015).
  * Bercovici (2015) D. Bercovici, in  _Treatise on Geophysics_ , Vol. 7, edited by G. Schubert (Elsevier, New York, 2015) 2nd ed., Chap. 1, pp. 1–42.
  * Aurnou  _et al._ (2020) J. M. Aurnou, S. Horn, and K. Julien, Phys. Rev. Res. 2, 043115 (2020).
  * Wells and Wettlaufer (2008) M. G. Wells and J. S. Wettlaufer, Geophys. Res. Lett. 35, L03501 (2008).
  * Ravichandran and Wettlaufer (2021) S. Ravichandran and J. S. Wettlaufer, J. Fluid Mech. 916, A28 (2021).
  * Boubnov and Golitsyn (1986) B. M. Boubnov and G. S. Golitsyn, J. Fluid Mech. 167, 503 (1986).
  * Vorobieff and Ecke (1998) P. Vorobieff and R. E. Ecke, Phys. Fluids 10, 2525 (1998).
  * Zhong  _et al._ (2010) J. Q. Zhong, M. D. Patterson, and J. S. Wettlaufer, Phys. Rev. Lett. 105, 5 (2010).
  * Ravichandran and Wettlaufer (2020) S. Ravichandran and J. S. Wettlaufer, J. Fluid Mech. 897, A24 (2020).
  * Ecke  _et al._ (1992) R. E. Ecke, F. Zhong, and E. Knobloch, Europhys. Lett. 19, 177 (1992).
  * Ning and Ecke (1993) L. Ning and R. E. Ecke, Phys. Rev. E 47, 3326 (1993).
  * Liu and Ecke (1997) Y. Liu and R. E. Ecke, Physical Review Letters 78, 4391 (1997).
  * Liu and Ecke (1999) Y. Liu and R. E. Ecke, Physical Review E 59, 4091 (1999).
  * Rossby (1969) H. T. Rossby, J. Fluid Mech. 36, 309 (1969).
  * Chandrasekhar (1953) S. Chandrasekhar, Proc. R. Soc. A 217, 306 (1953).
  * Veronis (1966) G. Veronis, J. Fluid Mech 24, 545 (1966).
  * Veronis (1968) G. Veronis, J. Fluid Mech. 31, 113 (1968).
  * Herrmann and Busse (1993) J. Herrmann and F. H. Busse, J. Fluid Mech. 255, 183 (1993).
  * Kuo and Cross (1993) E. Y. Kuo and M. C. Cross, Phys. Rev. E 47, R2245 (1993).
  * Ecke (2023) R. E. Ecke, Physica D 444, 133579 (2023).
  * Kunnen  _et al._ (2013) R. P. Kunnen, H. J. Clercx, and G. J. V. Heijst, J. Fluid Mech. 727, 509 (2013).
  * Ecke  _et al._ (2021) R. E. Ecke, X. Zhang, and O. Shishkina, Preprint, arXiv:2106.07710 (2021).
  * Goldstein  _et al._ (1993) H. F. Goldstein, E. Knobloch, I. Mercader, and M. Net, J. Fluid Mech. 248, 583 (1993).
  * Lopez  _et al._ (2007) J. M. Lopez, F. Marques, I. Mercader, and O. Batiste, Journal of Fluid Mechanics 590, 187 (2007), publisher: Cambridge University Press.
  * Ravichandran  _et al._ (2022) S. Ravichandran, S. Toppaladoddi, and J. S. Wettlaufer, J. Fluid Mech. 941, A39 (2022).
  * Horn and Schmid (2017) S. Horn and P. J. Schmid, J. Fluid Mech. 831, 182 (2017).
  * Lopez  _et al._ (2009) J. M. Lopez, F. Marques, A. M. Rubio, and M. Avila, Physics of Fluids 21, 114107 (2009), publisher: American Institute of Physics.
  * Favier and Knobloch (2020) B. Favier and E. Knobloch, J. Fluid Mech. 895, 1 (2020).


# Highlights from the Pierre Auger Observatory

a for the Pierre Auger Collaborationb
a Osservatorio Astrofisico di Torino (INAF) and INFN, Sezione di Torino, Italy
bObservatorio Pierre Auger, Av. San Martín Norte 304, 5613 Malargüe, Argentina
E-mail<EMAIL_ADDRESS>
Full author list: http://www.auger.org/archive/authors_icrc_2019.html

###### Abstract:

The Pierre Auger Observatory has collected data for more than 15 years,
accumulating the world’s largest exposure to ultrahigh energy cosmic rays
(UHECRs). The energy spectrum of the UHECRs has been measured up to
$10^{20.2}$ eV, and their mass composition has been studied exploiting both
the fluorescence and surface detectors of the Observatory. The analysis of the
UHECRs arrival directions allowed us to measure their large-scale anisotropies
over more than three decades in energy. The searches for intermediate-scale
anisotropies are being performed with an exposure now reaching approximately
$100,000$ $\mathrm{km^{2}~{}sr~{}yr}$. For the first time, the fluctuations in
the number of muons in inclined air showers above $4\times 10^{18}$ eV have
been obtained, while more information in the energy range
$10^{17.5}-10^{18.3}~{}{\rm eV}$ comes from the direct measure of muons with
the underground scintillators of the Observatory. Moreover, we discuss the
limits obtained on the fluxes of neutral particles and the results of the
searches for UHE neutrinos from transient astrophysical sources. The new
perspectives opened by the current results call for an upgrade of the
Observatory, whose main aim is the collection of new information about the
primary mass of the highest energy cosmic rays to understand the origin of the
observed flux suppression. The progress of the upgrade of the Observatory,
AugerPrime, is presented and discussed.

Figure 1: Left: layout of the Pierre Auger Observatory, showing the WCD (black
dots) and the azimuthal field of views of the 27 FD (blue and red lines). The
two laser facilities (CLF and XLF) for the monitoring of the aerosol content
in the atmosphere are indicated by the red dots, the AERA detectors in cyan.
Right: photograph of one of the (upgraded) WCD stations, with a FD site
visible in the background.

##  1 Introduction

The Pierre Auger Observatory was built more than 15 years ago thanks to the
contribution of more than 400 scientists from 17 different countries and to
the essential help of a team of on-site physicists and technicians, with the
primary scientific aim of studying cosmic rays in the ultrahigh energy range
above $\sim 10^{17}$ eV, to understand their nature and to find their sources.
Its layout is shown in the left panel of Fig.1. The surface detector (SD1500)
is formed by 1600 water Cherenkov stations (WCD) deployed in a 1.5 km grid
over a total area of about 3000 km2, and is used to measure the density and
the arrival time distribution of the shower particles at the ground. This area
is overlooked by four fluorescence detectors (FD), each consisting of six
telescopes with elevation angle from 1∘ to 30∘, for the measurement of the
energy deposit of the showers in the atmosphere. A photograph of one of the SD
stations is shown in the right panel of Fig.1. In the North-Western part of
the experimental site, 61 WCD (SD750) are deployed in a denser array with a
detector spacing of 750 m, together with three high-elevation fluorescence
telescopes (HEAT), with elevation angle from 30∘ to 60∘. This region is
exploited to widen the explorable energy region down to about $10^{16.5}$ eV.
The Auger Engineering Radio Array (AERA) complements the measurements of low
energy showers by detecting their radio emission, using more than 150 radio
antennas. Different atmospheric monitoring devices are employed to monitoring
the atmosphere. All details about the Observatory can be found in [1].

The rich harvest of the Pierre Auger Collaboration covers different and
complementary fields of research. The quest for the origin of UHE cosmic rays
relies on the measurement of the energy spectrum and mass composition of the
primaries, on multi-messenger studies and on extensive searches for anisotropy
at both large and intermediate angular scale; our most updated results are
discussed in Sec.2. The properties of multiparticle production are explored
thanks to the collection of different observables linked to the muonic and
electromagnetic components of the showers; the outcome is presented in Sec.3.
Our current efforts in upgrading the Observatory are summarized in Sec.4. Due
to the limited number of pages, only selected results are presented here; we
refer the reader to [59, 60, 61, 62, 63, 64, 65, 66] for further interesting
studies from the Pierre Auger Observatory presented at this conference. All
contributions are available within one volume in [67].

##  2 Astrophysics of UHE cosmic rays

###  2.1 Energy spectrum

A huge exposure of about 60,000 $\mathrm{km^{2}~{}sr~{}yr}$ allows us to
measure the cosmic ray energy spectrum, exploiting more than 215,000 events
collected by the SD1500 array with energies above $2.5\times 10^{18}$ eV and
arriving at zenith angles less than $60^{\circ}$ [2].
The reconstruction of these events is described in [3]. The signals from
through-going atmospheric muons, collected every minute, are used to convert
the particle densities in each SD station into a signal in units of Vertical
Equivalent Muons (VEM), providing a common reference to all stations. The
accuracy in the evaluation of the arrival direction of the shower depends on
the start times of the individual stations; it is derived by means of Monte
Carlo simulations based on the EPOS-LHC model, and at the highest energies it
results to be better than $1^{\circ}$ for all simulated angles and masses.
The energy estimator of the SD is chosen to be the signal evaluated at 1 km
from the core, where the fluctuations in the value of $S(1000)$ due to
uncertainties in the lateral distribution function are minimized. The
uncertainty on $S(1000)$ decreases from 15% to 5% for shower sizes increasing
from 10 to 250 VEM. By applying the Constant Intensity Cut (CIC) method,
$S(1000)$ is corrected for the attenuation of showers when propagating through
different amounts of atmosphere, normalizing it to the average arrival
direction of $38^{\circ}$ ($S_{38}$). We employ a new parametrization of the
CIC which takes into account the energy dependence of the shower attenuation
[2], due to changes in composition with increasing energy or, for fixed
primary mass, to the evolution of the electromagnetic fraction with respect to
the total signal.

Using the FD technique, we can measure the nearly calorimetric energy
deposited in the atmosphere by the electromagnetic component of the EAS.
Exploiting the correlation of the SD energy estimator $S_{38}$ with the FD
energy in hybrid events, i.e. those detected by both the SD and FD, we obtain
the energy scale of the Observatory and a calibration of all SD events in a
model-independent way. This correlation is shown in the left panel of Fig.2
for the SD1500 vertical sample, together with those measured with the SD750
and the SD1500 horizontal data sets. For the latter, a completely different
reconstruction is performed, yielding to the energy estimator $N_{19}$, the
relative muon content with respect to simulated proton showers at 10 EeV [4].
At this conference, we presented a review of the energy scale of the
Observatory, discussing the FD energy resolution and further checks on the
systematic uncertainties [5]. Real-time measurements of the atmospheric
conditions are mandatory for the correct evaluation of the fluorescence light
emitted by the showers. The main contribution to their systematic
uncertainties comes from the determination of the aerosol content; in
particular, we showed the importance of using hourly aerosols quantification
to avoid a significant worsening of the resolution on both the energy and the
$X_{\mathrm{max}}$ determination [6]. The reconstruction of the shower profile
has been improved by using a new function, where the shape parameters are free
from correlations, in this way allowing better-behaved fits [7]. The invisible
energy, i.e. that carried by particles that do not dissipate all their energy
in the atmosphere, is derived in a fully data-driven approach [8], as such
free from the systematic uncertainties affecting the hadronic interaction
models used in the simulations. Its current parametrization is based on the
horizontal events (muon dominated) and adds up to a fraction of 20 to 12% of
the total energy ranging from $10^{17}$ to $10^{20}$ eV, with systematic
uncertainties of less than 2%. The FD energy resolution amounts to about 8%,
adding the contributions coming from the atmosphere, the detector and
reconstruction and the invisible energy. We confirm the overall systematic
uncertainty of 14% on the energy scale [9].

The vertical energy spectrum is shown in the right panel of Fig.2 together
with those from the other independent and complementary data sets.

Figure 2: Left: Correlation between the FD energy and the SD energy estimators
as measured using the different data sets. Right: Energy spectra measured at
the Pierre Auger Observatory.

Figure 3: Left: Evolution of the spectral index as a function of energy. The
spectral indexes are obtained from power law fits to the spectrum over sliding
windows of 3 bins in log(E/eV). Right: The energy spectrum from the
combination of the different measurements.

At this conference, we presented for the first time the energy spectrum of
cosmic rays down to 100 PeV using the SD750 data [10], obtained by exploiting
a set of additional triggers allowing us to lower the threshold of full
efficiency to $10^{17}$ eV, thus approaching the region of the so-called ”2nd
knee”. Also in this case, the measurement is performed using a fully data-
driven approach. The spectrum below the ankle cannot be correctly described by
a single power law, which is excluded at the level of $4.1\sigma$. This
conclusion is confirmed by the measurement of the showers approaching HEAT
along the telescope axes, which, being dominated by the air-Cherenkov light
emission, have a lower detection threshold [11]. In this analysis, the energy
spectrum is derived between $10^{16.5}$ and $10^{18}$ eV with systematic
uncertainties dominated by those in the FD energy scale and in the exposure
evaluation (14% and 15% respectively).

The spectrum obtained by combining the five data sets exploiting the total
exposure of $\sim$80,000 $\mathrm{km^{2}~{}sr~{}yr}$ is shown in the right
panel of Fig.3. A functional form made by a sequence of different power laws
is used to fit it, as suggested by a study of the evolution of the spectral
slope assuming a spectrum $\mathrm{dN/dE\sim E^{-\gamma}}$ and fitting it in
sliding windows of three energy bins, as shown in the left panel of Fig.3. As
indicated in the plot, the spectrum is characterized by two clear inflection
points corresponding to the second knee around $10^{17}$ eV and to the ankle
near $6\times 10^{18}$ eV. A new feature is visible around $10^{19}$ eV, where
the spectral slope changes from $\gamma_{2}=(2.2\pm 0.2)$ to
$\gamma_{3}=(3.2\pm 0.1)$. We confirm the suppression of the flux above $\sim
5\times 10^{19}$ eV. The energy at which the integral spectrum drops by a
factor of two below what would be the expected with no steepening is
$E_{1/2}=14\pm 4$ EeV, at variance with that expected for uniformly
distributed sources of protons, as also indicated by the results from our
composition measurements.

###  2.2 Primary composition

The measurements of the depth of maximum development of the shower in the
atmosphere and of its fluctuations together with that of the primary energy
provide us with the most reliable information on the UHECRs composition.
Indeed, $\left<X_{\mathrm{max}}\right>$ is directly proportional to the
logarithmic mass of the primary particle, while its variance is a convolution
of the intrinsic shower-to-shower fluctuations and of the dispersion of masses
in the primary beam.
At this conference, we presented an update to this measurement based on more
than 47,000 hybrid events, more than 1000 of which above 10 EeV [12]. Above
$10^{17.8}$ eV, this corresponds to an increase of 20% in statistics, while
the lowest energy range is covered by the events collected by the HEAT
telescopes as presented in [13]. The evolution of
$\left<X_{\mathrm{max}}\right>$ and its standard deviation with energy is
shown in Fig.4.

Figure 4: The evolution of $\left<X_{\mathrm{max}}\right>$ (left) and
$\sigma(X_{\mathrm{max}})$ (right) with energy in comparison with the
predictions for proton and iron nuclei of the post-LHC hadronic models EPOS-
LHC, Sibyll 2.3c and QGSJetII-04.

We measure an elongation rate for $\left<X_{\mathrm{max}}\right>$ of $77\pm
2\textrm{(stat)}$ $\mathrm{g~{}cm^{-2}/decade}$ below $E_{0}=10^{18.32\pm
0.03}$ eV and $26\pm 2\textrm{(stat)}$ $\mathrm{g~{}cm^{-2}/decade}$ above
this energy. The result is in agreement with our previous findings [13, 14]
and points to a composition getting lighter up to $E_{0}$ and going towards
intermediate-heavy masses above it, at variance with the model expectations
for constant compositions, which suggest an elongation rate of $\sim 60$ g
cm-2/decade in the whole energy range. The evolution of
$\sigma(X_{\mathrm{max}})$ at the highest energies suggests a quite pure and
heavy composition, while in the lower energy range it is compatible with both
a light or mixed one.
Converting the two moments of the $X_{\mathrm{max}}$ distributions to those of
the logarithmic primary mass $\ln A$ as in [15], we concluded that the mass is
lightest at $E_{0}$, with $\ln A$ $\sim$ 0.8 and 1.4 for EPOS-LHC and
Sibyll2.3c respectively. The QGSJetII-04 model produces negative, unphysical,
values of $\sigma^{2}$($\ln A$), and as such it is unable to describe our data
correctly.

Besides measuring the particle densities at different distances from the core,
the SD allows us to obtain the distribution of the arrival times of particles
at each station. The dependence of the rise-time (i.e. the time needed for the
signal to go from 10 to 50% of its total value) on the distance of the shower
maximum to the ground and on the electromagnetic to muonic components of the
EAS was used in [16] to obtain information about the primary mass composition
from SD, thus exploiting its 100% duty cycle. At this conference, we presented
an update of that analysis [17] based on three more years of data collection
and an extended angular range of accepted events to $\theta<60^{\circ}$,
obtaining a data sample 30 times larger than that available from the FD above
3 EeV.
Exploiting the correlation of the SD observable towards $X_{\mathrm{max}}$, we
could derive the $\left<X_{\textrm{\tiny{max}}}^{SD}\right>$ evolution with
energy up to $E>10^{19.9}$ eV, although with large systematic uncertainties.
The results confirm the conclusion obtained in the analysis of the FD data.

Hadronic showers generated by heavy primary particles develop higher in the
atmosphere and contain more muons with respect to those produced by lighter
primaries. A study was published in [18] to derive information about the mass
spread in the ankle region by exploiting the correlation between
$X_{\mathrm{max}}$ and the signal in WCDs at 1000 meters from the core,
$S(1000)$ (which becomes more dominated by muons as the zenith angle
increases). Being based on hybrid events, this analysis is less prone to
systematic uncertainties and is not affected by our ignorance of the hadronic
interactions employed in the models. We presented at this conference an update
of the analysis [12], based on a statistics about two times larger with
respect to the published results, considering the energy range
$10^{18.5}-10^{19}~{}{\rm eV}$. To avoid a decorrelation due to the spreads of
energies and zenith angles, we rescaled $X_{\mathrm{max}}$ and $S(1000)$ to a
reference energy of 10 EeV and a zenith angle of $38^{\circ}$. The correlation
between the two observables is expected to be positive for constant
compositions in all models, while the experimental finding results in
$r_{G}=-0.069\pm 0.017$, with a systematic uncertainty $\Delta
r_{G}\textrm{(sys.)}=^{+0.01}_{-0.02}$. The dependence of the correlation
coefficients $r_{G}$ on $\sigma$(lnA) is shown in Fig.5, where the
experimental result is indicated by the shaded grey area.

Figure 5: Left: Dependence of the correlation coefficients $r_{G}$ on
$\sigma$(lnA). Right: evolution of the correlation coefficient for data (full
circles) in four different logarithmic energy bins. EPOS-LHC is here adopted
as hadronic interaction model to derive the predictions for the different
compositions.

The points correspond to the correlation expected in EPOS-LHC for different
mixtures of (p, He, O, Fe) nuclei, their colour to the corresponding value of
$\left<\textrm{lnA}\right>$. The simulations are compatible with data in the
region encompassed by the two vertical dashed lines. Similar results are
obtained if different models are employed; in all cases, the correlation hints
to a spread of masses in the range $0.85<\sigma\textrm{(lnA)}<1.6$. As shown
in the right panel of Fig.5, the correlation at energies
$10^{18.5}-10^{18.7}~{}{\rm eV}$, below the ankle, is significantly different
from zero (6.4$\sigma$), while smaller mixings are allowed above this range.
In the considered energy range $10^{18.5}-10^{19.0}~{}{\rm eV}$, pure
compositions and compositions consisting of only protons and helium can be
excluded at $5\sigma$ level.

An astrophysical interpretation of the Auger data on the energy spectrum and
mass composition can be obtained starting from different assumptions on the
possible sources. By assuming the same model of [19], in which the sources of
UHECRs are of extragalactic origin and accelerate nuclei in electromagnetic
processes with a rigidity-dependent maximum energy,
$E_{\textrm{max}}(Z)=E_{\textrm{max}}(p)/Z$, a preliminary comparison of the
energy spectrum and composition expected at the top of the Earth’s atmosphere
to the measured ones can be obtained in a combined fit procedure. As shown in
Fig.6, even using this simple model we reached the statistical power and the
knowledge of systematic uncertainties such to be able to reproduce the
different features of the spectrum and the average behaviour of the
$X_{\mathrm{max}}$ and its standard deviation. The best-fit parameters hint to
a quite hard injection spectrum and a low rigidity cut-off for protons, thus
implying a non-negligible fraction of heavier elements at the highest
energies. Within the systematic uncertainties, the results agree with those
previously published [19].

Figure 6: Left: The simulated energy spectrum (multiplied by $E^{3}$) at the
top of the Earth’s atmosphere obtained with the best fit parameters (see
text): all-particle (brown curve), $A=1$ (red), $2\leq A\leq 4$ (grey), $5\leq
A\leq 22$ (green), $23\leq A\leq 38$ (cyan), $A\geq 39$ (blue). The combined
energy spectrum as measured by Auger (Fig.3, right) is shown for comparison
with the black dots. Right: The first two moments of the $X_{\mathrm{max}}$
distributions as predicted for the model (brown curve) versus pure
compositions. Only the energy range indicated by the solid brown line is
included in the fit. The measured mean $X_{\mathrm{max}}^{SD}$ are shown with
purple triangles for comparison.

###  2.3 Multi-messengers

With its high potential in the identification of photons and neutrinos, the
Pierre Auger Observatory has a key role in the field of multi-messenger
astrophysics. Cosmogenic neutrinos and photons are produced in the
interactions of UHECRs in the background radiation fields during their
propagation from their sources to Earth. Neither are deflected in the Galactic
or intergalactic magnetic fields; they point back to their sources, making
them ideal messengers in targeted searches. While photons cover travel
distances of the order of $\sim 4.5$ Mpc at 1 EeV, neutrinos allow us to probe
sources up to cosmological distances. Their fluxes depend on the properties of
the sources and on the composition of the primary beam. A copious production
of $\nu$ and $\gamma$ can be expected, e.g., if the suppression of the UHECR
flux above a few tens of EeV is due to propagation effects and if the proton
component is dominant. On the contrary, an explanation of the cutoff as due to
the exhaustion of the sources would lead to much lower fluxes of neutral
particles and would point to a mixed composition.
The selection of photons in Auger is based on the fact that photon-induced
showers present a more elongated profile in the atmosphere, and thus a larger
$X_{\mathrm{max}}$, a steeper lateral distribution, causing the involvement of
a lower number of SD stations in the events, and a reduced production of muons
with respect to hadronic showers [20].
Neutrinos are looked for based on the selection of horizontal showers. In
hadronic-induced showers above $\sim 60^{\circ}$, the electromagnetic
component is indeed almost completely absorbed in the atmosphere, and only
muons are detected in the SD. On the contrary, in the case of neutrino events
with similar arrival directions, the first interaction would happen lower in
the atmosphere, producing a considerable amount of electrons and photons at
the ground. Two main categories of events are considered: Earth-skimming,
induced by $\nu_{\tau}$ travelling from below the Earth crust in directions
between $90^{\circ}$ and $95^{\circ}$ and producing a $\tau$ lepton which can
then generate a shower above the SD, and downward-going events due to
neutrinos of any flavour. Thanks to the larger grammage and to the density of
the Earth with respect to the atmosphere, the first is the most sensitive
channel [21].
The current upper limits on the cosmogenic fluxes of photons and neutrinos are
shown in Fig.7.

The energy range covered in the search for diffuse photons is here extended to
about three decades, thanks to an exposure now reaching $40,000$
$\mathrm{km^{2}~{}sr~{}yr}$, and to the analysis of data from the low energy
enhancements of the Observatory, namely SD750 and HEAT. The limits above EeV
energies put strong constraints on top-down models of UHECRs origin [22] and
on the most optimistic models of photon production from the interaction of
protons with the CMB.

Figure 7: Upper limits on the cosmogenic photon (left) and neutrino (right)
fluxes compared to limits from other experiments and to model predictions
([20, 21] and refs.therein).

The single flavour 90% upper limit on the integrated neutrino flux, for an
assumed flux $\sim E_{\nu}^{-2}$ and no neutrino candidates, is
$k_{90}<4.4\times 10^{-9}$ $\mathrm{GeV~{}cm^{-2}~{}sr^{-1}~{}s^{-1}}$ in the
energy range $10^{17}-2.5\times 10^{19}$ eV. The differential upper limit
characterizes the energy dependence of the sensitivity of a neutrino
experiment: as shown in the right panel of Fig.7, with the Auger Observatory
the best sensitivity is achieved for energies around 1 EeV. Describing the
evolution of the sources by $\Psi(z)\propto(1+z)^{m}$, up to a maximum
redshift $z_{\textrm{max}}$, we could perform a scan in the
$(z_{\textrm{max}},m)$ phase space and obtain the expected flux and the number
of cosmogenic neutrino events in Auger (using the propagation code CRPropa
[23]) for each configuration. From the non-observation of neutrinos, we could
thus exclude a significant part of the parameter space, as shown in the left
panel of Fig.8.

Besides diffuse neutrino fluxes, exploiting the very good angular resolution
of the Observatory (less than $0.5^{\circ}$ for $\theta>60^{\circ}$) we can
look for steady sources of UHE neutrinos. Upper bounds on the normalization of
a differential neutrino flux $E_{\nu}^{-2}$ as a function

Figure 8: Left: Constraints on UHECR source evolution models (see text).
Right: Upper limits at 90% C.L. on the normalization $k_{PS}$ of a single
flavour point-like flux of UHE neutrinos $dN/dE=k_{PS}E^{-2}$ as a function of
the source declination $\delta$.

of declination are shown in the right panel of Fig.8. The energy range of
Auger is different and complementary to those of IceCube and Antares [24, 25].

At this conference, we also presented the results of our studies on transient
sources of UHE neutrinos of the order of an hour or less. The sensitivity of
the SD to these kind of sources depends not only on the declination, as in the
case of steady sources, but also on the efficiency of the detection during the
time occurrence of the transit. For sources in transit within the field of
view corresponding to the zenith angle range $\theta=(60^{\circ}-95^{\circ})$,
where the SD is most sensitive to UHE neutrinos, the sensitivity is unrivaled
with respect to other neutrino detectors.
At the time of the GW170817, detected by LIGO and Virgo [26], the source was
located in the most sensitive part of the Auger field of view, and we provided
the strongest constraints on the UHE neutrino fluence from the merger [27].
Two other types of potential transient sources were scrutinized in [28].
Binary black hole mergers (BBH), detected by the gravitational wave detectors,
could in principle be sources of neutrinos, if matter debris and strong
magnetic fields are present in the source environment [29]. Combining searches
for neutrinos from 21 BBH mergers, and considering an emission spectrum
$E^{-2}$, the non-observation of any candidates in the 24 hours after the
mergers brought to an upper limit on the luminosity of a universal isotropic
source of UHE neutrinos of $L_{\textrm{up}}<4.3\times 10^{46}$ erg/s, with
maximum sensitivity about 14.6 hours after the merger.
UHE neutrino events from the blazar TXS 0506+056, for which a correlated
emission of neutrinos and photons was found [30, 31] below the energy
threshold of Auger, were also searched for in two different time windows
correlated to the neutrino excess found with IceCube and to the following
associated high-energy neutrino alert, but no UHE neutrinos have been
observed. Despite the smaller sensitivity to neutrinos from the TXS 0506+056
flares with respect to IceCube, due to the periodic visibility with
interruptions, we underline that we are scrutinizing a different higher energy
range, unreachable by other observatories.

###  2.4 Arrival directions

In [32], the Pierre Auger Collaboration announced the observation of a large-
scale anisotropy in the arrival directions of cosmic rays above $8\times
10^{18}$ eV. At this conference, this analysis was extended to cover more than
three decades in energy, exploiting about 15 years of data [33]. For energies
above 4 EeV, data from the fully efficient SD1500 array were used, reaching a
total exposure of $92,500$ and $60,700$ $\mathrm{km^{2}~{}sr~{}yr}$ for events
with $\theta\leq 80^{\circ}$ and $\theta\leq 60^{\circ}$ respectively.

Under the assumption that higher multipoles are negligible, we found a total
dipolar amplitude for $E\geq 8$ EeV of $d=0.066\pm 0.012$, pointing $\sim
125^{\circ}$ away from the direction of the Galactic center, as such
indicating an extragalactic origin of the modulation. The overall flux in this
energy bin is shown in the left panel of Fig.9.

Figure 9: Left: The CR flux above 8 EeV, averaged on top-hat windows of
$45^{\circ}$ radius (equatorial coordinates). The Galactic plane and the
Galactic center are indicated by a dashed line and a star respectively. Right:
Energy dependence of the dipolar amplitude measured in four energy bins above
4 EeV.

The possible presence of quadrupolar components has been considered, but their
amplitudes result to be negligible, and even in this case the dipole is
consistent with that found by neglecting higher-order multipoles.
The same study, free from assumptions on the shape of the underlying angular
distribution, has been performed considering a full-sky search for large-scale
anisotropies, in a joint working group of the Pierre Auger and Telescope Array
collaborations [34]. By a careful reweighting of the exposures and cross-
calibrating the respective energy scales, the data from the two experiments
could be analyzed together. The dipolar amplitude found is again consistent
with that obtained by Auger alone, with smaller uncertainties when allowing
for non-vanishing quadrupole moments.

Thanks to the large exposure, with the Auger Observatory we studied the
evolution of the dipolar amplitude with energy, which is shown in the right
panel of Fig.9. The linear growth can be reproduced with $d=d_{10}(E/10\
\textrm{EeV})^{\beta}$, where $d_{10}=0.051\pm 0.007$ and $\beta=0.96\pm
0.16$; an evolution of the amplitude constant with energy is disfavoured at
the level of $5.1\sigma$ by a likelihood ratio test. A study of the effect on
the extragalactic dipole of the deflections caused by the Galactic magnetic
field showed that in this case, depending on the particle rigidities, the
dipole direction would tend to align in the direction of the spiral arms and
suffer a reduction of the amplitude [35].

Further studies have been conducted to extend the search for large-scale
anisotropies to a lower energy range, exploiting the data from the infill
array SD750 and using the East-West method [36], less sensitive to systematic
effects that could result in spurious anisotropies, in place of the standard
Rayleigh analysis. The amplitudes and phases of the equatorial dipole in the
extended energy range are shown in Fig.10. There, we also indicate with the
99% CL upper bounds the result in the bins where the measured amplitude is
smaller than the value within which 99% of the simulations with isotropic
distributions of the same number of events would fall. Although the amplitudes
at low energy are not significant, the phase of the equatorial dipole appears
to change from values pointing close to the right ascension of the Galactic
center below EeV energies towards the opposite direction, again indicating an
extragalactic origin for the dipolar anisotropy above a few EeV.

At higher energies, where the rigidities of the particles are high enough and
cosmic rays from far sources are suppressed due to the energy losses they
suffered during propagation, searches for intermediate-scale anisotropies have
been performed by exploiting 15 years of data, reaching an unprecedented
exposure of $\sim 100,000$ $\mathrm{km^{2}~{}sr~{}yr}$ in a declination range
between $-90^{\circ}$ and $+45^{\circ}$ [37].

Figure 10: Amplitude (left) and phase (right) of the equatorial dipole
amplitude in the extended energy range. The upper limits $d_{\perp}^{UL}$ are
indicated by arrows for the bins where $d_{\perp}<d_{\perp}^{99\%}$ (see
text). The results at lower energy from IceCube, IceTop and KASCADE-Grande
([33] and refs. therein) are also shown.

A model-independent blind search for overdensities was performed over the
whole field of view, in a wide energy threshold range $E_{\mathrm{thr}}=32-80$
EeV using a 1 EeV step and scanning circular regions $\Psi$ from $1^{\circ}$
to $30^{\circ}$ in steps of $1^{\circ}$. The most significant excess was found
for $E>38$ EeV in an angular window of $27^{\circ}$ centered in the direction
$(\alpha=202^{\circ},\delta=-45^{\circ})$, a region densely populated by
extragalactic objects among which there is the closest radio-Galaxy Centaurus
A. Indeed, the excess of events in its direction, already pointed out in the
past [38], reached a one-sided post-trial significance of $3.9\sigma$ for
energies above $E_{\mathrm{thr}}=37$ EeV and $\Psi=38^{\circ}$.

The arrival directions of UHECRs were also studied against the distribution of
nearby extragalactic matter traced by different catalogs of possible sources
up to a distance of $\sim 250$ Mpc. Similarly to [39], we considered (a) a
sample of 32 starburst Galaxies (SBGs) selected based on their continuum
emission at 1.4 GHz, used as a proxy for their UHECR flux; (b) a selection of
$\gamma$AGNs from the 3FHL catalog, weighting the sources with the integral
flux from 10 GeV to 1 TeV; (c) the 2MRS catalog of nearby matter farther than
1 Mpc; (d) a selection of Swift-BAT radio-loud and radio-quiet AGNs.

Figure 11: Left: Maximum likelihood-ratio as a function of energy threshold
for the models based on SBGs and $\gamma$AGNs. The results are shown in the
attenuation (full line) and no-attenuation (dashed line) scenarios. Right:
Cumulated test statistics for $E_{\mathrm{thr}}=38$ EeV as a function of the
time ordered number of events (for the SBG-only model). The number of events
at the time of [39] and of this conference are indicated by the red arrows.

In this analysis, we exploited not only the source directions but also the
possible ansatzes for the relative UHECR fluxes from source candidates
provided by the observations from Fermi-LAT. Furthermore, our current
knowledge of the cosmic ray composition could be used to evaluate the
attenuation in intensity due to the energy losses during propagation.
A likelihood ratio analysis allowed us to evaluate the smearing angle and the
fraction of anisotropic cosmic rays for each case. The test statistics
obtained for the correlation with SBGs and $\gamma$AGNs is shown in the left
panel of Fig.11. The highest significance is obtained for the case of SBGs,
where we found an anisotropic fraction of $(11\pm 5)\%$ of events with energy
above 38 EeV in a smearing angle of $15^{\circ}$. Note that the attenuation of
particles in the case of the nearby SBGs is much less important that for more
distant blazars. After penalization for the scanning in energy, the
significance of the correlation results to be $4.5\sigma$. The cumulated test
statistics for $E_{\mathrm{thr}}=38$ EeV is shown in the right panel of Fig.11
as a function of the number of collected events.

##  3 Hadronic interactions

The interpretation of the experimental observables in terms of primary
composition is prone to systematic uncertainties, mainly due to the lack of
knowledge on hadronic interactions at ultra-high energies. On the one hand,
additional data from collider and fixed-target experiments are needed to lower
these uncertainties. On the other hand, the interactions of primary cosmic
rays in the atmosphere can be exploited to study the hadronic interaction
models in a kinematic and energy region not accessible by human-made
accelerators. Indeed, exploiting Auger data, we reach center-of-mass energies
up to $\sqrt{s}\sim 400$ TeV, more than 30 times those attainable at LHC and
explore interactions in the very forward region of phase space on targets of
$\left<A\right>\sim 14$.
The shower development depends on many different features of the hadronic
interactions. In particular, by collecting the electromagnetic radiation
emitted by the shower particles crossing the atmosphere and its depth of
maximum development $X_{\mathrm{max}}$, we get information about the first
interaction cross section [40]. The measure of the muonic component at the
ground is more sensitive to the details of the hadronic interactions along
many steps of the cascade, like the multiplicity of the secondaries and the
fraction of electromagnetic component with respect to the total signal. On the
contrary, the intrinsic muon fluctuations mostly depend on the first
interaction [41].
Clear evidence for a deficit of the number of muons predicted by the models
was reported by our Collaboration by exploiting inclined showers, where the
electromagnetic component has been fully absorbed by the atmosphere [42] and
in the hadronic component [43]. The study of the muon production depth [44]
and of the time profiles of the signals recorded with SD [16] further proved
the inability of models to describe all the components of the showers
correctly.

Figure 12: Mean logarithmic muon density $\left<\textrm{ln}\rho_{35}\right>$vs
$\left<X_{\mathrm{max}}\right>$ expected by showers simulated at two different
energies with EPOS-LHC and QGSJetII-04, in comparison to the experimental
results from UMD and FD.

Figure 13: Average relative muon number (left) and shower-to-shower
fluctuations (right) in inclined events.

At this conference, we showed that the deficit of muons in the models is well
visible also at lower energies, by directly measuring the muon content of EAS
with an engineering array of underground muon detectors (UMD) deployed in the
infill area [45]. In Fig.12, the correlation of the muon densities measured by
the UMD with the $\left<X_{\mathrm{max}}\right>$ measured by the FD is shown
at two different energies ($10^{17.5}$ and $10^{18}$ eV).
The mean logarithmic muon densities are linearly dependent on the mean
logarithmic primary mass, so that they can be related to the
$\left<X_{\mathrm{max}}\right>$ in proton and iron simulations. The resulting
predicted muon content is 38% too low at both energies when EPOS-LHC is used,
the mismatch increasing to 50% to 53% at the two energies in the case of
QGSJetII-04.

Moreover, we presented for the first time the measurement of the shower-to-
shower fluctuations of the muonic component, at the same time updating the
results on the mean muon number exploiting hybrid inclined showers [46]. They
are shown in Fig.13; note that systematic errors (square brackets) are
dominant for $\left<R_{\mu}\right>$, while for the muon fluctuations the
uncertainties are mainly statistical (error bars). The expectations for the
relative fluctuations are compatible with the experimental results, while
models and data show a significant discrepancy in the average muon scale. The
results thus suggest that models are describing reasonably well the
distribution of energy going into the electromagnetic component after the
first interaction, while the discrepancy in the overall muon number should be
explained by resorting to changes of different hadronic interaction
characteristics along all the shower stages.

Non-standard mechanisms, like departures from relativity such as Lorentz
invariance violation (LIV), can result in a perturbative modification of the
particle dispersion relation
$E_{i}^{2}=p_{i}^{2}+m_{i}^{2}+\sum_{n=0}^{\infty}\delta^{(n)}_{i}E_{i}^{n+2}$,
(where $i$=type of particle, $n$=approximation order). The UHECRs collected at
the Pierre Auger Observatory, coming from extragalactic sources, represent the
ideal candidates to search for these effects, as these are expected to be
suppressed at low energies and for short travel distances. At this conference,
we presented searches for Lorentz invariance violation effects in three
different sectors [47], by modifying the propagation of hadrons or photons or
by considering the changes which would appear in the evolution of the showers
in the atmosphere, and comparing the predictions on the energy spectrum and
composition with those measured, in a combined fit procedure. In the first
case, our data is better described with a low maximum rigidity at sources,
which leads to a poor sensitivity to LIV effects. In the photon sector, the
limits of $\delta^{(1)}_{\gamma}\sim 10^{-40}$ eV-1 and
$\delta^{(2)}_{\gamma}\sim 10^{-60}$ eV-1 obtained in our analysis are two and
four orders of magnitude more restrictive than those in [48] for $n$=1 and
$n$=2 respectively.

##  4 Conclusion and future steps

The rich harvest of results from the Pierre Auger Observatory presented at
this conference is based on an unprecedented exposure and on a more and more
careful understanding of our data and their systematic uncertainties. The
large-scale dipolar modulation above 8 EeV is confirmed, and its amplitude is
found growing with energy. The extension of the measurement of the amplitude
and phase of the modulation to more than three orders of magnitude allowed us
to confirm a Galactic origin for cosmic rays below the EeV, while the
direction of the dipole strongly support the conclusion that they are
extragalactic above few EeV. Blind searches for intermediate and small-scale
anisotropies confirmed the existence of an excess in the direction of
Centaurus A. A growing significance (now above 4.5$\sigma$) was found for the
correlation among the directions of cosmic rays with $E>38$ EeV and the
position of starburst galaxies in a catalog including objects like NGC4945 and
M83 in the CenA region, but also NGC253 near the Southern Galactic pole. More
studies including the effects of magnetic fields and larger catalogs are
required in the future to confirm these findings.

The measurement of the energy spectrum of cosmic rays covers a wide range of
energies, from $\sim 10^{16.5}$ to above $10^{20}$ eV, and it is derived
without assumptions on models or composition. Two inflection points are
clearly visible corresponding to the second knee and to the ankle, a new
feature around $10^{19}$ eV is now appearing thanks to the much reduced
statistical errors. The region around the ankle appears to be dominated by a
mixed composition, with a pure or too light one excluded with a significance
of $\sim 5\sigma$ by our measurements based on hybrid events. The best
information on the primary mass relies on the evaluation of the
$X_{\mathrm{max}}$ distributions and their moments, hinting to a composition
which gets heavier above $2\times 10^{18}$ eV, a conclusion confirmed by new
measurements exploiting the 100% duty cycle of the SD. These results point to
an origin for the confirmed steepening of the flux above $\sim 5\times
10^{19}$ eV more related to the exhaustion of the sources rather than to
propagation effects. Information about the mass composition in the suppression
region is however still missing.

New results on hadronic interaction models at ultrahigh energy come from the
study of muons: a good description of the energy sharing after the first
interaction is confirmed by the first-ever measurement of the evolution of the
muon intrinsic fluctuations in extensive air showers. The deficit in the
expected total number of muons is now confirmed also at energies down to
$10^{17.5}$ eV thanks to the measurements performed with the UMD, and most
probably due to effects related to the subsequent steps of the cascades in the
atmosphere.

The AugerPrime upgrade of the Pierre Auger Observatory [49] is being built to
address the open questions raised by the current results. The interpretation
of the suppression in the flux by differentiating between a cut-off due to
propagation effects and the maximum energy reached in the sources can provide
fundamental constraints on the sources of the ultrahigh energy cosmic rays and
their properties. To set the stage for future experiments and assess the
feasibility of charged particle astronomy, it is mandatory to evaluate the
possible existence of a fraction of protons at the highest energies. New
information about hadronic interactions in a range of energy and in a
kinematic region which is not accessible in human-made accelerators will allow
us to break the degeneracy between primary mass and hadronic interactions and
furthermore to search for non-standard mechanisms. These questions must be
addressed not only by reaching larger exposures but more importantly by
providing new information on the primary mass composition at ultra-high
energies on an event-by-event basis.

Each station of the SD will be equipped by a 3.8 m2 slab of plastic
scintillator on top (the Surface Scintillator Detector, SSD [50]), new SD
electronics and an additional small photomultiplier inside the WCD for the
extension of the dynamic range [51, 52]. Besides the scintillator, a radio
antenna (RD) will be mounted above the station for the detection of the radio
signals from the extensive air showers in the frequency range from 30 to 80
MHz[53]. Additionally, the UMD are going to be deployed beside each WCD in the
infill region of the Observatory, to provide direct muon measurements [54].

The most direct technique to evaluate the muon component exploits the
different responses of the WCD and SSD at station level, relating their total
signals to the electromagnetic and muonic fluxes at the ground. Indeed, the
ratio of the electromagnetic to muonic components of the showers is more than
a factor of two higher in a thin scintillator with respect to the WCD over a
very wide range of distances to the core.
This information can be obtained employing a matrix inversion method [55],
although in this case the measured muon signals are subject to large
fluctuations, being restricted to individual stations. A more general approach
based on the properties of universality of air showers [56] or multivariate
analyses would allow us to take advantage of all the information provided by
AugerPrime, not only by correlating the detector signals at different
distances to the core but also considering their temporal structure (arrival
times of the particles at the stations). In this case, an event-by-event
reconstruction will provide the energy, the depth of maximum shower
development in the atmosphere and the number of muons. Moreover, the
implementation of the radio detector will allow us to collect a new different
kind of hybrid events: the combination of the WCD and RD will be used to
exploit the correlation of the electromagnetic energy (from radio) and the
number of muons (from the WCD) for horizontal air showers. Indeed, the size of
the radio footprint exceeds tens of km2 for very inclined showers
($\theta=60^{\circ}-84^{\circ}$) [57], with dozens of stations involved.

Since 2016, 12 fully upgraded stations (the Engineering Array) are running in
the field with new electronics, higher sampling, larger dynamic range.
Furthermore, more than 350 upgraded stations (WCD+SSD) have already been
deployed at the time of this conference. By exploiting the Engineering Array
and a pre-production array of 80 upgraded stations running in the field with
the old electronics, we could check the performances and calibration of the
new detectors [58]. The signals in the WCD are measured up to saturation
($\sim 650$ VEM) by the large PMTs. In the superposition region and above the
large PMT saturation, they are derived exploiting the small PMT. The measured
correlation among the signals of the WCD and SSD demonstrated the validity of
the independent calibration procedures and showed that the required dynamic
range in the upgraded SD is nicely covered up to the highest particle
densities ($\sim 20,000$ VEM). The differences of response of the two
detectors were studied and confirmed by evaluating the dependency of the
signal ratio with zenith, energy and distance to the core.

The prototypes of the RD are now installed in the field for in-situ evaluation
of their performances, the starting of the mass production is foreseen for the
end of 2019.

The production and deployment of the AugerPrime detectors and electronics will
be completed by 2020, for a data-taking planned up to 2025 and beyond.

## References

  * [1] The Pierre Auger Coll., Nucl. Instrum. Meth. A 798 (2015) 172.
  * [2] V.Verzi [Pierre Auger Coll.], PoS(ICRC2019) 450.
  * [3] D. Mockler [Pierre Auger Coll.], PoS(ICRC2019) 353.
  * [4] The Pierre Auger Coll., JCAP 08 (2014) 019.
  * [5] B. Dawson [Pierre Auger Coll.], PoS(ICRC2019) 231.
  * [6] V.M. Harvey [Pierre Auger Coll.], PoS(ICRC2019) 283.
  * [7] The Pierre Auger Coll., JCAP 03 (2019) 18.
  * [8] The Pierre Auger Coll., Phys. Rev. D (in press) (2019) [arXiv:1901.08040].
  * [9] V. Verzi [Pierre Auger Coll.], Proc. 33th ICRC (2013) [arXiv:1307.5059].
  * [10] A.Coleman [Pierre Auger Coll.], PoS(ICRC2019) 225.
  * [11] V.Novotny [Pierre Auger Coll.], PoS(ICRC2019) 374.
  * [12] A.Yushkov [Pierre Auger Coll.], PoS(ICRC2019) 482.
  * [13] J. Bellido [Pierre Auger Coll.], PoS(ICRC2017) 506
  * [14] The Pierre Auger Coll., Phys.Rev. D90 (2014), 12, 122005.
  * [15] The Pierre Auger Coll., JCAP 1302 (2013) 026.
  * [16] The Pierre Auger Coll., Phys. Rev. D 96 (2017) 122003.
  * [17] C.J.Todero Peixoto [Pierre Auger Coll.], PoS(ICRC2019) 440.
  * [18] The Pierre Auger Coll., Phys. Lett. B762 (2016) 288.
  * [19] The Pierre Auger Coll., JCAP04 (2017) 038.
  * [20] J.Rautenberg [Pierre Auger Coll.], PoS(ICRC2019) 398.
  * [21] F.Pedreira [Pierre Auger Coll.], PoS(ICRC2019) 979.
  * [22] The Pierre Auger Coll., JCAP04 (2017) 009.
  * [23] R.Alves Batista et al.,JCAP05 (2016) 038.
  * [24] The IceCube Coll., Astrophys.J. 835 (2017) 151.
  * [25] The ANTARES Coll., Phys.Rev. D96 (2017) 082001.
  * [26] B.P.Abbot et al., Astrophys.J.Lett. 848 (2017) L12.
  * [27] A.Albert et al., Astrophys.J.Lett. 850 (2017) L35.
  * [28] M.Schimp [Pierre Auger Coll.], PoS(ICRC2019) 415.
  * [29] K.Kotera K, J. Silk, Astrophys. J. 823 (2016) L29.
  * [30] The IceCube Coll.,Science 361(2018) 1378.
  * [31] The IceCube Coll.,Science 361(2018)147.
  * [32] The Pierre Auger Coll., Science 357 (2017) , 1266.
  * [33] E. Roulet [Pierre Auger Coll.], PoS(ICRC2019) 408.
  * [34] A. di Matteo [Pierre Auger and Telescope Array Coll.], PoS(ICRC2019) 439.
  * [35] The Pierre Auger Coll., Astrophys.J. 868 (2018) 4.
  * [36] R.Bonino et al., Astrophys.J. 738 (2011) 67.
  * [37] L. Caccianiga [Pierre Auger Coll.], PoS(ICRC2019) 206.
  * [38] The Pierre Auger Coll., Astrophys. J. 804 (2015) 15.
  * [39] The Pierre Auger Coll., Astrop.J.Lett., 853 (2018) L29.
  * [40] The Pierre Auger Coll., Phys.rev.Lett. 109 (2012) 062002.
  * [41] L.Cazon et al., Phys. Lett. B 784 (2018) 68.
  * [42] The Pierre Auger Coll., Phys. Rev. D91 (2015) 032003.
  * [43] The Pierre Auger Coll., Phys. Rev. Lett. 117 (2016) 192001.
  * [44] The Pierre Auger Coll., Phys. Rev. D 90 (2014) 012012; D90 (2014) 039904(A); D92 (2015) 019903(E).
  * [45] F.Sanchez [Pierre Auger Coll.], PoS(ICRC2019) 411.
  * [46] F.Riehn [Pierre Auger Coll.], PoS(ICRC2019) 404.
  * [47] R.Guedes Lang [Pierre Auger Coll.], PoS(ICRC2019) 327.
  * [48] R.Guedes Lang et al., Astrophys.J. 853 (2018) 23.
  * [49] A.Castellina [Pierre Auger Coll.], EPJ Web Conf. 210 (2019) 06002.
  * [50] J.Pekala [Pierre Auger Coll.], PoS(ICRC2019) 380.
  * [51] D.Nitz [Pierre Auger Coll.], PoS(ICRC2019) 370.
  * [52] M.Bohacova [Pierre Auger Coll.], PoS(ICRC2019) 199.
  * [53] B.Pont [Pierre Auger Coll.], PoS(ICRC2019) 395.
  * [54] A.M.Botti [Pierre Auger Coll.], PoS(ICRC2019) 202.
  * [55] A. Letessier-Selvon et al., Nucl. Instrum. Meth. A767 (2014) 41.
  * [56] D.Maurel et al., Proc. 33th ICRC (2013) 600.
  * [57] The Pierre Auger Coll., JCAP 2018 (2018) 026.
  * [58] A.Taboada [Pierre Auger Coll.], PoS(ICRC2019) 434.
  * [59] M.Wirtz [Pierre Auger Coll.], PoS(ICRC2019) 469.
  * [60] R.Mussa [Pierre Auger Coll.], PoS(ICRC2019) 1131.
  * [61] M.Schimassek [Pierre Auger Coll.], PoS(ICRC2019) 1147.
  * [62] J.Glombitza [Pierre Auger Coll.], PoS(ICRC2019) 270.
  * [63] M.Gottowik [Pierre Auger Coll.], PoS(ICRC2019) 274.
  * [64] O.Deligny [Pierre Auger Coll.], PoS(ICRC2019) 234.
  * [65] A.Barbano [Pierre Auger Coll.], PoS(ICRC2019) 842.
  * [66] B.Garcia [Pierre Auger Coll.], PoS(ICRC2019) 265.
  * [67] The Pierre Auger Coll., arXiv:1909.09073 (2019).