# On the ice-nucleating potential of warm hydrometeors in mixed-phase clouds

Michael Krayer1, Agathe Chouippe1,a, Markus Uhlmann1, Jan Dušek2, Thomas
Leisner3
1Institute for Hydromechanics, Karlsruhe Institute of Technology (KIT),
Karlsruhe, Germany
2ICube, Fluid Mechanics Group, Université de Strasbourg, Strasbourg, France
3Institute of Meteorology and Climate Research, Atmospheric Aerosol Research
Department,
Karlsruhe Institute of Technology (KIT), Eggenstein-Leopoldshafen, Germany
anow at: ICube, Fluid Mechanics Group, Université de Strasbourg, Strasbourg,
France

###### Abstract

The question whether or not the presence of warm hydrometeors in clouds may
play a significant role in the nucleation of new ice particles has been
debated for several decades. While the early works of Fukuta and Lee, (1986)
and Baker, (1991) indicated that it might be irrelevant, the more recent study
of Prabhakaran et al., (2019) suggested otherwise. In this work, we are aiming
to quantify the ice-nucleating potential using high-fidelity flow simulation
techniques around a single hydrometeor and use favorable considerations to
upscale the effects to a collective of ice particles in clouds. While we find
that ice nucleation may be enhanced in the vicinity of a warm hydrometeor by
several orders of magnitude and that the affected volume of air is much larger
than previously estimated, it is very unlikely that this effect alone causes
the rapid enhancement of ice nucleation observed in some types of clouds,
mainly due to the low initial volumetric ice concentration. Nonetheless, it is
suggested to implement this effect into existing cloud models in order to
investigate second-order effects such as ice nucleus preactivation or
enhancement after the onset of glaciation.

##  1 Introduction

The formation of hydrometeors in clouds is of great importance for the
prediction of weather and cloud electrification, as well as for the
hydrological cycle, and thus, eventually for the evolution of climate.
However, despite its relevance and great research effort over the past
decades, many aspects remain poorly understood. One such puzzle is the
discrepancy between the concentration of ice particles and that of available
ice nuclei in airborne observations by several orders of magnitude (Pruppacher
and Klett,, 2010) which has been observed for various cloud types (Koenig,,
1963; Auer et al.,, 1969; Hobbs,, 1969; Hobbs and Rangno,, 1985; Mossop,,
1985; Hogan et al.,, 2002). This phenomenon has been termed _ice enhancement_
and various mechanisms which amplify primary ice nucleation have been proposed
to explain the observed surplus in ice particles (Field et al.,, 2016) .

The most promising class of enhancement mechanisms is the so-called secondary
ice production (SIP), where new ice is formed from preexisting ice particles.
Commonly accepted SIP include rime-splintering (Hallett and Mossop,, 1974),
fragmentation of ice (Vardiman,, 1978; Takahashi et al.,, 1995; Bacon et al.,,
1998) and freezing drops (Hobbs and Alkezweeny,, 1968). Most of these
mechanisms have been implemented into cloud models with explicit ice
microphysics (see Field et al., (2016) for a recent overview), which, however,
are not capable of satisfactorily explaining the large amount of ice particles
in observations. Further mechanisms have been proposed in the past whose
relative importance is still to be evaluated.

While studying the effect of supersaturation on primary ice formation, Gagin,
(1972) proposed several mechanisms which locally produce high values of
supersaturation, and thus, regions of significantly enhanced nucleation
activity. He suggested that freezing hydrometeors, which attain higher
temperatures than the surrounding air due to the release of latent heat, may
cause transient supersaturation by simulateneous evaporation and heat transfer
and linked it to the observation of satellite drops which have previously been
observed experimentally during the freezing of supercooled droplets by Dye and
Hobbs, (1968). This hypothesis was corroborated by Nix and Fukuta, (1974), who
investigated numerically the transient freezing process of an isolated
droplet.

Another phenomenon which leads to localized supersaturation around
hydrometeors is the riming process of ice particles in mixed-phase clouds with
high liquid-water content (Gagin,, 1972). Here, sufficiently large ice
particles collect supercooled droplets from their surroundings, which then
accumulate on its surface and subsequently freeze, leading to similar non-
equilibrium conditions as observed for the freezing drop. The supersaturation
around a riming ice particle has been investigated numerically by Fukuta and
Lee, (1986) assuming steady conditions. Indeed, they found that the air in the
vicinity of a warm ice particle can be highly supersaturated, while the
magnitude and spatial extent strongly increase with increasing difference in
surface to ambient temperature. However, even at low cloud temperatures, the
supersaturated regions do not extend far away from the hydrometeor in their
simulations, which led to the conclusion of Rangno and Hobbs, (1991) that the
overall ice enhancement is likely to be negligible.

While the previously mentioned studies focused on the quantification of the
supersaturation field, Baker, (1991) attempted to quantify the actual effect
on ice enhancement around warm hydrometeors. He concluded that, although
significant ice enhancement factors appear to be possible, the affected air
mass seems to be too small to substantially contribute to the explanation of
the discrepancy between ice crystal and ice nucleus concentrations.

Recently, the potential of hail and rain to nucleate droplets has gained new
attention when Prabhakaran et al., (2017) observed condensation in the wake of
cold droplets in a moist convection apparatus. In their experiment, a
pressurized mixture of sulfur hexafluoride/helium was used and operating
conditions were chosen such that slight supersaturations led to homogeneous
condensation, which could indeed be observed in the wake of larger falling
droplets. However, their results can only be qualitatively transfered to
Earth’s atmosphere mainly due to the difference in primary nucleation
mechanism. Moreover, the mechanism to create supersaturation is different from
the previously discussed one in the sense that the droplets are colder than
the surrounding air.

To overcome these shortcomings, Prabhakaran et al., (2019) conducted a similar
laboratory experiment using moist air which has been seeded with aerosol
particles as an operating medium to investigate heterogeneous nucleation in
the wake of a warm falling droplet. Again, significant nucleation was observed
in the wake of the falling hydrometeor. While the extent of wake-induced
nucleation has not been quantified, the affected volume of air appears to be
significantly larger than the predictions of Fukuta and Lee, (1986) and Baker,
(1991) were suggesting. This discrepancy might be related to the strong
assumptions on the flow made in these early works, which neglect important
features of the wake of falling objects that are nowadays accessible to
numerical simulations (Johnson and Patel,, 1999; Bouchet et al.,, 2006; Zhou
and Dušek,, 2015). In particular, it has been shown that both unsteadiness and
vortical structures of the flow strongly affect the temperature and vapor
concentration in the vicinity and far downstream of a falling sphere (Bagchi
et al.,, 2001; de Stadler et al.,, 2014), and thus strongly affect the
distribution of supersaturation around hydrometeors (Chouippe et al.,, 2019).

In Chouippe et al., (2019), we presented a framework for high-fidelity
numerical simulations of heat and mass transfer around a falling ice particle
which is not in thermal equilibrium with its surroundings. Even though the
focus was predominantly set towards several methodological questions, it was
shown qualitatively that the local supersaturation differs strongly from the
simpler considerations of Fukuta and Lee, (1986).

The present work aims to revisit the details of the supersaturation field
around an idealized falling hydrometeor and to quantify the volume of air
which is affected by the presence of hydrometeors. Furthermore, it is yet to
be evaluated how significant meteor-induced enhancement of ice nucleation is
in clouds (Korolev et al.,, 2020), and therefore, we attempt to link our
results to heterogeneous nucleation of aerosol particles.

##  2 Methodology

In order to assess the spatial structure of supersaturation around
hydrometeors, knowledge on the flow around it is essential. This is not an
easy task, since the features of the flow may strongly depend on parameters
such as the size, shape and surface properties of the hydrometeor, which
typically vary substantially, especially if ice particles are concerned.

In this work we utilize the numerical framework previously presented in
Chouippe et al., (2019) and model the hydrometeor as a sphere with constant
diameter $D$, constant surface temperature $T_{p}$ and a constant vapor
pressure $e_{v,p}$ at its surface. The latter is determined by the assumption
of local equilibrium at the surface, and thus, corresponds to the saturation
vapor pressure $e_{sat,j}$ at $T_{p}$. Here, $j$ denotes the phase of water on
the surface of the hydrometeor, where subscript ”$i$” will be used to denote
the solid phase and ”$w$” to denote the liquid phase.

The boundary conditions for the heat and mass transfer problem are motivated
by the conditions expected for riming ice particles in the wet-growth regime.
A surface temperature of $T_{p}=0\,$\mathrm{\SIUnitSymbolCelsius}$$ is
assumed, i.e. the surface of the ice sphere is warmer than the local cloud
temperature, since the latter is generally below the freezing point in mixed-
phase clouds. Please note that at this surface temperature the equilibrium
vapor pressure with respect to both the liquid and solid phase coincide, and
hence, no assumptions on the phase at the surface have to be made. Since the
actual riming process is not modelled explicitly, but only enters through the
boundary conditions of the heat and mass transfer problem, the results of this
work are more general and can in principle be applied to any configuration
which contains a mechanism leading to a warm hydrometeor, such as a freezing
droplet or rapid decreases in cloud temperature.

One of the key parameters for the investigation of wake-induced ice nucleation
is the distribution of the saturation ratio

$S_{j}(e,T)=e/e_{sat,j}(T),$ (1)

where $e$ denotes the local vapor pressure. A parametrization of $e_{sat,j}$
with respect to temperature for both phase equilibria in the temperature range
of interest is adopted from Murphy and Koop, (2005). The spatial distribution
of $S_{j}$ can then be reconstructed from the temperature field $T$ and the
vapor concentration $n_{v}$, which are governed by the transport equations

$\displaystyle\frac{\partial T}{\partial t}+\vec{u}\cdot\vec{\nabla}T$
$\displaystyle=$ $\displaystyle\mathcal{D}_{T}\vec{\nabla}^{2}T,$ (2)
$\displaystyle\frac{\partial n_{v}}{\partial t}+\vec{u}\cdot\vec{\nabla}n_{v}$
$\displaystyle=$ $\displaystyle\mathcal{D}_{n_{v}}\vec{\nabla}^{2}n_{v},$ (3)

where $\mathcal{D}_{T}$ and $\mathcal{D}_{n_{v}}$ denote the heat and vapor
diffusivities and $\vec{u}=(u,v,w)^{T}$ is the velocity field of the
surrounding air. The partial pressure of water is linked to these transported
quantities by

$e=n_{v}/k_{b}T,$ (4)

where $k_{b}$ is Boltzmann’s constant. Even though the diffusivities of both
vapor and heat are similar in magnitude, it was shown by Chouippe et al.,
(2019) that assuming equal diffusivities leads to an underestimation of the
saturation ratio, and hence, both equations will be treated separately.

Under the boundary conditions described earlier, Eq. (2) leads to an outward
heat flux, because the surface temperature of the hydrometeor is higher than
that of its surrounding. Furthermore, at
$T_{p}=0\,$\mathrm{\SIUnitSymbolCelsius}$$ the vapor concentration at the
meteor’s surface is generally higher than in the ambient, since latter is
assumed to be in equilibrium with the liquid phase at the corresponding cloud
temperature $T_{\infty}<T_{p}$ due to the presence of droplets in mixed-phase
clouds, i.e. $e_{\infty}=e_{sat,w}(T_{\infty})$. This results in an outward
vapor flux, and hence, the hydrometeor will also evaporate. It is therefore of
primary importance to resolve the temporal and spatial variations of both
temperature and concentration fields.

In Chouippe et al., (2019) we have shown that buoyant forces due to density
variations within the fluid phase caused by variations in temperature and
concentration are negligible at the parameter point of interest. The flow can
then be approximated by the incompressible Navier-Stokes equations,

$\displaystyle\vec{\nabla}\cdot{\vec{u}}$ $\displaystyle=$ $\displaystyle 0,$
(5) $\displaystyle\frac{\partial\vec{u}}{\partial
t}+(\vec{u}\cdot\vec{\nabla})\vec{u}$ $\displaystyle=$
$\displaystyle-\frac{1}{\rho_{f}}\vec{\nabla}p+\nu\vec{\nabla}^{2}\vec{u},$
(6)

where $p$ denotes the hydrodynamic pressure, $\rho_{f}$ the density and $\nu$
the kinematic viscosity of the fluid. The flow is driven by the falling motion
of the hydrometer, which is assumed to be moving with a constant velocity
$\vec{v}_{p}$ equal to its terminal velocity $v_{T}$, i.e.

$\vec{v}_{p}=-v_{T}\vec{e}_{z},$ (7)

where $\vec{e}_{z}$ is the unit vector in the $z$-direction. The fluid ahead
of the hydrometeor is assumed to be at rest. This setup is equivalent to a
system with a fixed sphere positioned in an upcoming flow, which differs from
that of a freely falling mobile sphere in the sense that fluctuations in
$\vec{v}_{p}$ are not permitted. The necessity to account for these variations
has been discussed in Chouippe et al., (2019) and it was found that it has
little influence in the present context.

Figure 1:  Spectral-element mesh in the axial-radial plane, where the lines
depict the boundaries of the $463$ elements. Each element contains ${[}6\times
6{]}$ collocation points. The three-dimensionality is introduced by Fourier
expansion in the azimuthal direction, which is truncated at the 7th Fourier
mode, resulting in a cylindrical domain. A uniform velocity profile with
constant temperature and vapor content is imposed at the upstream boundary,
while the downstream boundary is subject to zero-gradient BCs for both
velocity and scalars. The lateral boundaries are stress-free at zero pressure
with zero-gradient BCs for the scalars. The surface of the spherical particle,
whose center is located at the origin of the coordinate system, is impermeable
and a no-slip BC for velocity is imposed as well as constant Dirichlet BCs for
the scalar fields.

The Navier-Stokes and scalar transport equations are solved numerically in
non-dimensional form. In particular lengths are scaled by the particle
diameter $D$, velocity components by the terminal velocity $v_{T}$ and the
scalar transport equations are formulated in terms of

$\displaystyle\tilde{T}$ $\displaystyle=(T-T_{\infty})/(T_{p}-T_{\infty}),$
(8) $\displaystyle\tilde{n}_{v}$
$\displaystyle=(n_{v}-n_{v,\infty})/(n_{v,p}-n_{v,\infty}),$ (9)

which has the advantage, that various dimensional boundary conditions, e.g.
various values of $T_{\infty}$, can be studied in post-processing from a
single simulation run. Under the simplifications stated, the non-dimensional
problem can be parametrized by the Reynolds number,
$Re=\left|\vec{v}_{p}\right|D/\nu$, the Prandtl number,
$Pr=\nu/\mathcal{D}_{T}$, and the Schmidt number,
$Sc=\nu/\mathcal{D}_{n_{v}}$. While the Reynolds number is varied up to a
value of $Re=600$, the Prandtl and Schmidt numbers are set, respectively, to
values of $Pr=0.72$ and $Sc=0.63$ in this study, which correspond to
representative values expected for humid air in the temperature range of
interest.

The numerical method employed is based on the method of Jenny and Dušek,
(2004), which has been used by e.g. Kotouč et al., (2008) and Chouippe et al.,
(2019) to simulate heat and mass transfer around spherical particles. It
relies on a spectral/spectral-element discretiztion of the Navier-Stokes
equations (Eq. 5,6) coupled to the transport of heat and mass (Eq. 2,3). The
spherical particle is placed at the origin of a cylindrical domain $\Omega$.
On the surface of the particle impermeability and no-slip boundary conditions
are imposed, while the scalar fields are subject to the constant Dirichlet
boundary conditions $\tilde{T}=\tilde{n}_{v}=1$. At the upstream boundary of
the simulation domain, a uniform velocity condition and
$\tilde{T}=\tilde{n}_{v}=0$ are enforced, while the downstream boundary is
subject to zero-gradient boundary conditions for velocity as well as the
scalars. The lateral boundaries are stress-free with zero-gradient boundary
conditions for the scalars and zero pressure is imposed. The axial-radial
plane is decomposed using the spectral element method of Patera, (1984), while
the homogeneous azimuthal direction is treated using Fourier decomposition.
The numerical mesh used in this work is shown in fig. 1. For the temporal
integration a time splitting method is used which consists of an explicit
third order Adams-Bashforth discretization for the advective terms and a first
order fully implicit discretization of the diffusion terms (Rønquist,, 1988).

In the vertical direction the domain has a total length of $62D$, where the
inflow length is $12D$ and the outflow length $50D$ measured from the center
of the sphere. The diameter of the domain is $16D$. The computational domain
has therefore been extended in the rear of the sphere compared to our former
simulations presented in Chouippe et al., (2019). In total, 463 two-
dimensional elements, each containing ${[}6\times 6{]}$ collocation points,
have been distributed over the domain. The azimuthal Fourier series is
truncated at the 7th mode. This resolution is comparable to our previous work
and has been shown to give good results for the momentum, as well has heat and
mass transfer. All relevant scales of the flow and the scalar fields are
resolved.

For more details on the numerical framework, the reader is referred to the
previous work of Chouippe et al., (2019).

##  3 Results

###  3.1 Hydrometeor wake regimes in clouds

The dynamics of the flow around spherical objects are, under the assumptions
stated in the previous section, fully parametrized by the Reynolds number,
which depends on the diameter and indirectly, through modulation of the
terminal velocity, on the density of the hydrometeor. Depending on the
Reynolds number value, various flow states emerge in the wake (Johnson and
Patel,, 1999; Jenny et al.,, 2004; Kotouč et al.,, 2009). At low values the
wake is axisymmetric and steady. When the first critical value of the Reynolds
number, $Re_{c,1}\approx 212$, is exceeded, the wake becomes oblique with
respect to the falling direction (_steady oblique regime)_ and only planar
symmetry may be observed (Ghidersa and Dušek,, 2000). At $Re_{c,2}\approx 273$
a second instability of Hopf type occurs (Ghidersa and Dušek,, 2000), which
leads to unsteady periodic vortex shedding, while planar symmetry is still
maintained (_oscillating oblique regime_). Eventually the vortex shedding
becomes chaotic at $Re_{c,3}\approx 360$ (Ormières et al.,, 1999) and all
instantaneous symmetries are lost (_chaotic regime_). The listed regimes
differ significantly in their ability to transfer heat and mass (Chouippe et
al.,, 2019), and thus, different characteristics in producing local
supersaturation are to be expected.

In order to estimate the distribution of ice particle wake regimes in clouds,
we adopt the size distribution of Marshall and Palmer, (1948),

$N=N_{0}\exp\mathinner{\left(-\lambda D\right)},$ (10)

where $N$ is the number concentration density of hydrometeors per unit volume
of air and $N_{0},\lambda$ are model parameters. Equation (10) has originally
been developed for raindrops, however, its validity for sufficiently large ice
particles has been demonstrated by Passarelli, (1978); Houze et al., (1979);
Gordon and Marwitz, (1984); Herzegh and Hobbs, (1985); Patade et al., (2015)
for various types of natural clouds. The values of the model parameters
$N_{0},\,\lambda$ are usually obtained by airborne measurements and given as
functions of the cloud temperature. One such parametrization is provided by
Houze et al., (1979) for frontal clouds within a temperature range from
$-42$\mathrm{\SIUnitSymbolCelsius}$$ to $+6$\mathrm{\SIUnitSymbolCelsius}$$,
which will be used in the following for the size distribution of primary ice
and is given by the constitutive equations

$\displaystyle N_{0}(T)$ $\displaystyle=5.5\cdot
10^{6}\exp\left(-0.09T\right)${\mathrm{m}}^{-4}$,$ (11)
$\displaystyle\lambda(T)$ $\displaystyle=9.6\cdot
10^{-1}\exp\left(-0.056T\right)${\mathrm{mm}}^{-1}$,$ (12)

where $T$ denotes the cloud temperature in $\mathrm{\SIUnitSymbolCelsius}$.
For this cloud type, ice enhancement has been previously reported in the
presence of riming ice particles (Hogan et al.,, 2002). Even though it is
known that the number density of ice particles of sub-millimeter size might
deviate significantly from Eq. (10) in a sub-exponential or super-exponential
manner (Passarelli,, 1978), depending on cloud conditions, we assume that Eq.
(10) holds nonetheless for all ice particle sizes for the sake of simplicity.

For reasons explained in section 3.3, our quantity of interest is not the
number concentration, but rather the volume fraction $\phi$ of hydrometeors in
clouds, which can be obtained by integrating the corresponding moment of the
Marshall-Palmer distribution, i.e.

$\phi=\int_{0}^{\infty}\frac{\pi D^{3}}{6}N_{0}\exp\mathinner{\left(-\lambda
D\right)}\operatorname{d\\!}D.$ (13)

In order to determine the wake regimes from the size distribution, the
terminal velocity needs to be approximated as a function of the ice particle
diameter, and hence, further assumptions on the density of the ice particles
have to be made. For graupel particles, the density may take values which
range between $0.05\,$\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$$ and
$0.89\,$\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$$, depending on growth
conditions and history (Pruppacher and Klett,, 2010). For our estimation we
choose a value of $0.6\,$\mathrm{g}\text{\,}{\mathrm{cm}}^{-3}$$, as we are
primarily concerned with ice particles in the wet-growth mode, which typically
constitute the upper end of the density range. Using this value and the
empirical drag law of Schiller and Naumann, (1933), the critical diameters for
regime transition are calculated from the critical Reynolds numbers. The total
solid volume fraction can then be subdivided into the contributions by ice
particles with a certain wake regime, i.e.

$\phi\equiv\sum_{j}\phi_{j}=\sum_{j}\int_{D_{j,\min}}^{D_{j,\max}}\frac{\pi
D^{3}}{6}N_{0}\exp\mathinner{\left(-\lambda D\right)}\operatorname{d\\!}D,$
(14)

where the index $j$ denotes one of the four wake regimes (steady axisymmetric;
steady oblique; oscillating oblique; chaotic) and $D_{j,\min}$, $D_{j,\max}$
the corresponding lower/upper limit for the hydrometeor diameter to be in this
regime.

Figure 2:  Volume fraction of ice particles in clouds as a function of ambient
temperature. The total volume fraction $\phi$ is displayed by a solid black
line (), while the contribution by the regimes is given by the colored lines.
Linestyles: axisymmetric regime (), steady oblique regime (), oscillating
oblique regime (), chaotic regime ().

Figure 2 shows the estimated volume fraction of frozen hydrometeors in clouds
as a function of cloud temperature. The values are typically smaller than
$10^{-5}$ and decrease exponentially with decreasing temperature. When
comparing the various regimes, it can be observed that the largest
contributions generally stem from hydrometeors in the axisymmetric or chaotic
regime. This becomes especially clear when looking at the relative
contribution of $\phi_{j}$ to the total volume fraction $\phi$ as shown in
fig. 3. As can be seen, more than $80$\mathrm{\char 37\relax}$$ of ice
particles (by volume) exhibit one of these two wake regimes. Chaotic wakes are
dominant at temperatures close to the freezing point, while axisymmetric wakes
dominate when the temperature is very low. We will therefore focus solely on
the axisymmetric and chaotic regimes in the following.

Figure 3:  Relative contribution to the total volume fraction by hydrometeors
in a certain regime. Linestyles: axisymmetric regime (), steady oblique regime
(), oscillating oblique regime (), chaotic regime ().

###  3.2 Supersaturation in the wake of hydrometeors

We now examine the saturation profiles in the wake of hydrometeors in the two
regimes of interest. Figure 4 shows isocontours of supersaturation w.r.t. ice,
defined as $s_{i}=S_{i}-1$, for two different values of the Reynolds number,
namely $Re=75$, which lies within the axisymmetric regime, and $Re=600$ in the
chaotic wake regime. As ambient temperature, a value of
$T_{\infty}=-30$\mathrm{\SIUnitSymbolCelsius}$$ was adopted, because this
roughly corresponds to the cloud temperature where the two regimes are of
equal significance (cf. fig. 3). Since the ambient fluid is already
supersaturated w.r.t. ice, the threshold of the isocontours is given as an
excess to the value in the ambient and we define the excess supersaturation as

$\tilde{s}\equiv s-s_{\infty}.$ (15)

Figure 4:  Isosurfaces of supersaturation in the wake at
$T_{\infty}=-30$\mathrm{\SIUnitSymbolCelsius}$$. The value of the isocontour
is $\tilde{s}^{*}_{i}=0.1$, i.e. ten percentage points higher than the ambient
supersaturation. Two different wake regimes are depicted, which correspond to
two different hydrometeor sizes in our framework. (a) axisymmetric regime at
$Re=75$, (b) chaotic regime at $Re=600$.

At $Re=75$, the flow is steady, and thus, so is the supersaturation field. In
the chaotic regime, the flow is characterized by time-dependent vortex
shedding from the ice particle’s boundary layer, and therefore excess
supersaturation appears intermittently. In both regimes, significant excess
supersaturation w.r.t. the ambient can be observed far downstream and the
volume of air which is affected by the wake is by far larger than the volume
of the meteor. This can be seen more clearly when averaging the fields in the
azimuthal direction, which is statistically homogeneous for both cases, as
well as in time for the unsteady flow. The averaged excess supersaturation
w.r.t. ice is shown in fig. 5 for both cases. The saturation profiles differ
substantially from the ones obtained by Fukuta and Lee, (1986) for similar
boundary conditions and Reynolds number. This is presumably due to their
strong simplification of potential flow, which is incapable of reproducing the
boundary layer and the flow in the wake of the ice particle correctly. This
indeed leads to strong modifications in the distribution of supersaturated
regions. Regions of high vapor content and relatively low temperatures, i.e.
supersaturated regions, which are created in the mixing layer close to the
riming particle are transported further downstream by the detaching vortices
in the chaotic regime. This effect does not occur in the simulations of Fukuta
and Lee, (1986), who observed a complete decay of supersaturation to the
inflow value within a distance of approximately $2.5D$ downstream of the ice
particle for $T_{\infty}=-30$\mathrm{\SIUnitSymbolCelsius}$$ while fig. 5
clearly indicates that regions with $\tilde{s}_{i}>0$ may be observed at
distances more than $50D$ from the rear at this temperature.

Figure 5:  Contours of excess supersaturation in the wake, averaged over time
and azimuthal direction at $T_{\infty}=-30$\mathrm{\SIUnitSymbolCelsius}$$.
(a) axisymmetric regime at $Re=75$, (b) chaotic regime at $Re=600$.

In order to compare the induced supersaturation for different values of
$T_{\infty}$ and in between different regimes, we compute the time-averaged
volume of air which is supersaturated above a given threshold,

$V_{s}(s^{*})\equiv\frac{1}{\tau}\int_{0}^{\tau}\int_{\Omega}H(s(\vec{x},t)-s^{*})\operatorname{d\\!}\Omega\operatorname{d\\!}t,$
(16)

where $V_{s}$ is the superaturated volume, $s^{*}$ the corresponding threshold
and $H$ the Heaviside step function. Figure 6 shows this quantity normalized
by the volume of the ice particle as a function of the threshold. It can be
seen, that in the axisymmetric regime, the supersaturated volume is generally
larger than in the chaotic regime. This is caused by the enhanced mixing
properties of the vortical wake structures in the latter case, which lead to a
faster decay of the scalar field.

For both regimes, the range of observed values of supersaturation is similar.
In fact, differences in range may only occur due to the distinct diffusivities
of the temperature and water vapor fields, which indeed lead to subsaturations
in the chaotic regime in some parts of the flow. While the range can be
estimated reasonably well using simple mixing arguments (Chouippe et al.,,
2019), we observe that the highest values of supersaturation only occur in a
very confined portion of the wake. However, if the temperature difference
between the ambient and the ice particle is sufficiently high, a strong excess
in supersaturation compared to the ambient value may occur in a volume
comparable to the size of the meteor, if no mechanism for depletion is
considered. This excess may be as high as 270% at a temperature difference of
$40$\mathrm{K}$$. For all values of $T_{\infty}$ the volume of air which is
affected by the wake-induced supersaturation is of the order of $10^{3}$
particle volumes.

Figure 6:  Volume of air where supersaturation exceeds a given threshold as a
function of the threshold. The volume is normalized by the volume of the ice
particle and three different temperatures are shown:
$T_{\infty}=-20$\mathrm{\SIUnitSymbolCelsius}$$ (),
$T_{\infty}=-30$\mathrm{\SIUnitSymbolCelsius}$$ (),
$T_{\infty}=-40$\mathrm{\SIUnitSymbolCelsius}$$ (). Solid lines correspond to
$Re=600$ (chaotic regime), while dashed lines show the data obtained for
$Re=75$ (axisymmetric regime).

###  3.3 Ice enhancement due to meteor wakes

Our spatially resolved data allows us, for the first time to our knowledge, to
quantify the significance of wake-induced supersaturation on ice enhancement.
Thus, in this section, we are now aiming to estimate how the nucleation of ice
is affected by this excess in supersaturation. We therefore adopt the commonly
used ice enhancement factor (e.g. Baker,, 1991), which is defined as the ratio
between the observed number concentration of ice nuclei (IN) and that expected
for a reference state. For our problem, it will be defined as

$f_{i}\equiv\frac{N_{IN}\,\text{(with meteors)}}{N_{IN}\,\text{(without
meteors)}}.$ (17)

The IN number concentration $N_{IN}$, i.e. those aerosol particles which can
be activated at a given supersaturation, is commonly approximated by a power-
law in terms of the supersaturation w.r.t. ice,

$N_{IN}=Cs_{i}^{\alpha}$ (18)

where $C,\alpha$ are constants depending on the aerosol composition (Huffman,,
1973). For natural aerosols, the exponent $\alpha$ typically ranges between
$3$ and $8$, where larger values are attributed to more polluted air masses,
e.g. near metropolitan or industrial areas (Pruppacher and Klett,, 2010). We
are aware that Eq. (18) is a very simplified approach to the problem and its
validility might be debatable. In particular, it does not take into account
the various characteristics of the different freezing modes, which are
dependent on the temperature and supersaturation w.r.t. the liquid phase, nor
the time scales of freezing, which might play an significant role in the
microphysical view. The results should therefore only be regared as a first
crude estimation on the importance of wake-induced ice enhancement in clouds.

Using Eq. (18), the local ice enhancement factor can be expressed as

$f_{i}(\vec{x})=\frac{\left<s_{i}^{\alpha}\right>_{t}(\vec{x})}{s_{i,\infty}^{\alpha}}.$
(19)

This quantity gives a quantitative measure on how much more ice is produced in
the wake of the hydrometeor compared to the ambient under the limitations
stated above. Figure 7 shows how the ice enhancement is distributed in the
wake of a hydrometeor at $T_{\infty}=-30$\mathrm{\SIUnitSymbolCelsius}$$ for
an exponent of $\alpha=8$. At this parameter point, which can be considered as
rather favorable, we observe that $f_{i}$ can locally reach extremely high
values of the order of $10^{4}$ in the near vicinity of the hydrometeor and
significant values greater than $10$ can still be observed in the far wake.

Figure 7:  Contours of local ice enhancement factor in the wake, averaged over
time and azimuthal direction at
$T_{\infty}=-30$\mathrm{\SIUnitSymbolCelsius}$$. The contour lines are spaced
logarithmically. (a) axisymmetric regime at $Re=75$, (b) chaotic regime at
$Re=600$.

Using Eq. (19), the supersaturated volume can be expressed as a function of
the ice enhancement factor in order to gain insight into the volumetric
distribution of $f_{i}$. Figure 8 shows the distributions for $\alpha=8$ and
various temperatures in both wake regimes. The ice enhancement is generally
higher in the axisymmetric regime than in the chaotic regime. Please note,
that for a more precise analysis of this behavior, the actual flow-driven
distribution of aerosol particles in the wake should be considered. Depending
on the flow dynamics, ice nucleating particles might accumulate in the wake or
be expelled from it (Homann and Bec,, 2015), which might have a non-negligible
impact on ice enhancement. However, the consideration of aerosol dynamics is
outside the scope of the present work.

Figure 8:  Volume of air with supersaturation above a given threshold as a
function of the ice enhancement factor at $\alpha=8$. The volume is normalized
by the volume of the ice particle and three different temperatures are shown:
$T_{\infty}=-20$\mathrm{\SIUnitSymbolCelsius}$$ (),
$T_{\infty}=-30$\mathrm{\SIUnitSymbolCelsius}$$ (),
$T_{\infty}=-40$\mathrm{\SIUnitSymbolCelsius}$$ (). Solid lines correspond to
$Re=600$ (chaotic regime), while dashed lines show the data obtained for
$Re=75$ (axisymmetric regime).

We therefore conclude that the wake of a hydrometeor may act as a site of
increased nucleation activity, and that the volume in which this increased
activity occurs is much larger than the volume of the hydrometeor itself. This
conclusion is consistent with recent experimental results of Prabhakaran et
al., (2019) who observed nucleation of water droplets and ice particles in the
wake of a hot drop under cold conditions with a temperature difference
comparable to the one presented in fig. 7.

In order to assess the significance of the wake effect in clouds, a global ice
enhancement factor needs to be computed, which takes into account the spatial
distribution of $f_{i}$ obtained by the numerical simulations, as well as the
volumetric ice concentration. We define a control volume $\mathcal{V}$ within
a cloud which is sufficiently large such that the hydrometeor size
distribution follows Eq. (10), but sufficiently small such that the saturation
field is uniform (with a value of $s_{i,\infty}$) if no hydrometeor is
present. The volume-averaged ice enhancement within this control volume is
given by

$\left<{f_{i}}\right>_{\mathcal{V}}=\frac{\int_{\mathcal{V}}\mathinner{\left(f_{i}(\vec{x})-1\right)}\operatorname{d\\!}\mathcal{V}}{\int_{\mathcal{V}}\operatorname{d\\!}\mathcal{V}}+1,$
(20)

where the integral has been decomposed into an excess contribution, which
evaluates to zero outside of the wake, and a base contribution, which is equal
to unity. At sufficiently small volume fractions, the wakes of individual
hydrometeors can be assumed to not interact, and hence, the integral in the
enumerator of the first term in Eq. (20) can be expressed as a sum of
contributions from independent hydrometeors, i.e.

$\int_{\mathcal{V}}\mathinner{\left(f_{i}(\vec{x})-1\right)}\operatorname{d\\!}\mathcal{V}=\sum_{(k)}\int_{\Omega^{(k)}}\mathinner{\left(f_{i}(\vec{x})-1\right)}\operatorname{d\\!}\Omega^{(k)},$
(21)

where $\Omega^{(k)}\subseteq\mathcal{V}$ denotes the simulation domain scaled
by the size of particle $(k)$. By defining the excess ice enhancement per
meteor volume as

$\tilde{f}_{i,\Omega^{(k)}}\equiv\frac{6}{\pi
D^{3}}\int_{\Omega^{(k)}}\mathinner{\left(f_{i}(\vec{x})-1\right)}\operatorname{d\\!}\Omega^{(k)}$
(22)

and assuming a continuous particle size distribution following Eq. (10), the
volume-averaged ice enhancement within $\mathcal{V}$ can be expressed as

$\left<f_{i}\right>_{\mathcal{V}}=1+\int_{0}^{\infty}\tilde{f}_{i,\Omega}\frac{\pi
D^{3}}{6}N_{0}\exp\mathinner{\left(-\lambda D\right)}\operatorname{d\\!}D.$
(23)

Since $\tilde{f}_{i,\Omega}$ is normalized by the volume of the ice particle,
its value only depends on the flow field in non-dimensional form, i.e. the
diameter dependency only enters through the value of the Reynolds number. For
reasons of simplicity, we assume that $\tilde{f}_{i,\Omega}$ can be
approximated for a given regime by the value determined for a single Reynolds
number within that regime, which allows us to rewrite Eq. (23) in the
simplified form

$\left<f_{i}\right>_{\mathcal{V}}=1+\sum_{j}\phi_{j}\tilde{f}_{i,\Omega_{j}},$
(24)

where $\phi_{j}$ is the volume fraction of meteors in regime $j$, as defined
in Eq. (14), and $\tilde{f}_{i,\Omega_{j}}$ the value of the excess ice
enhancement evaluated for a single value of the Reynolds number in that
regime. As has already been discussed and shown in fig. 3, the axisymmetric
and chaotic regimes contribute the most to the volume fraction. We therefore
disregard the contribution of the two other regimes and adjust the threshold
of regime transition accordingly.

$\left<f_{i}\right>_{\mathcal{V}}=1+\phi_{axi}\tilde{f}_{i,\Omega_{axi}}+\phi_{cha}\tilde{f}_{i,\Omega_{cha}}$
(25)

Both $\phi_{j}$ and $\tilde{f}_{i,\Omega}$ are functions of ambient
temperature, and thus the global ice enhancement factor originating from
meteor wakes can be expressed as a function of cloud temperature. While the
former decreases exponentially with decreasing temperature, the latter
exhibits a strong increase leading to a counteracting effect. Furthermore, a
large number of ice particles pertain to the axisymmetric regime for low
temperatures, which is more favorable in terms of ice enhancement.

Please be aware that even though our aim is to quantify the wake-induced ice
formation, the volume fraction of ice will be regarded as constant. The reason
for this is that the approach used in this study does not allow us to derive
the time scales of nucleation nor growth of newly created ice, and thus, the
time-dependent coupling with the size distribution is inaccessible. Therefore,
the following considerations merely apply to the initial state of a possible
rapid glaciation process.

Figure 9:  Global ice enhancement factor as a function of cloud temperature
for various values of $\alpha$. The inset shows the same data, but in semi-
logarithmic scale. Linestyles: $\alpha=3$ ( ), $\alpha=4$ ( ), $\alpha=5$ ( ),
$\alpha=6$ ( ), $\alpha=7$ ( ), $\alpha=8$ ( ).

Figure 9 shows the global ice enhancement factor as a function of ambient
temperature for different values of $\alpha$, the exponent in Eq. (19). It can
be seen that significant ice enhancement in the global sense only occurs at
cloud temperatures below $-30$\mathrm{\SIUnitSymbolCelsius}$$ in conjunction
with high values of $\alpha$. Even then, the enhancement is only marginal with
$\left<f_{i}\right>_{\mathcal{V}}$ being smaller than $8$. We therefore
conclude that under typical cloud conditions hydrometeors do not directly lead
to substantially enhanced ice nucleation on a global scale.

Since the small volume fraction of ice is the limiting factor for ice
enhancement on a global scale, we revisited our assumption on the size
distribution of ice particles in order to assess the sensitivity to the
parametrization. Using the parametrization of Heymsfield et al., (2002) for
deep subtropical and tropical clouds, the ice volume fraction may be up to
five times larger at $T_{\infty}=-40$\mathrm{\SIUnitSymbolCelsius}$$. However,
at higher cloud temperatures, i.e. when the temperature difference between ice
particles and the ambient is moderate, the volume fraction is of the same
order of magnitude as for the presented results, such that the conclusions
which have been drawn for frontal clouds also persist for this type of cloud.

##  4 Conclusion

In this study we have performed numerical simulations of momentum, heat and
mass transfer around a warm hydrometeor in order to assess the distribution of
supersaturation in its wake, and moreover, the implications on ice nucleation
enhancement. Our simulation method is based on a body-conforming
spectral/spectral-element discretization and all relevant scales of the flow
problem have been resolved. The hydrometeor is assumed to be of spherical
shape and to possess a uniform surface temperature of
$0\,$\mathrm{\SIUnitSymbolCelsius}$$, while the ambient temperature has been
varied in a range between $-40\,$\mathrm{\SIUnitSymbolCelsius}$$ and
$0\,$\mathrm{\SIUnitSymbolCelsius}$$. The vapor concentration is kept at
saturation value with respect to ice at the particle surface, while it is
saturated with respect to water in the ambient (reflecting the presence of
supercooled droplets). Two different values of the Reynolds number have been
simulated in order to capture the characteristics of the most relevant wake
regimes, namely $Re=75$, where the wake is steady and axisymmetric, and
$Re=600$, where the wake is chaotic.

We found that significant values of supersaturation can be attained in the
wake of warm hydrometeors, which persist long enough to be observed at more
than 50 particle diameters downstream of the meteor for sufficiently high
differences in temperature. The supersaturated volume of air exceeds the
estimations by Fukuta and Lee, (1986) by far, which is attributed to the more
accurate representation of the flow in the current study. This is an important
observation since one of the key arguments for disregarding wake-induced ice
nucleation is the proclaimed small zone of influence (Baker,, 1991).

Using a power-law approximation for the ice enhancement factor, we estimate
that heterogeneous nucleation may be locally enhanced by a factor comparable
to the discrepancy between ice nuclei and ice particles observed in field
measurements. This enhancement increases strongly with decreasing cloud
temperature and so does the affected volume of air, which is typically of the
order of several hundred particle diameters. However, due to the low
volumetric concentration of ice in clouds, this local effect alone unlikely
serves as an explanation for the global discrepancy, as arguments on upscaling
have shown.

Nonetheless, it is conceivable that the present mechanism in conjunction with
one (or several) secondary ice processes is of greater relevance to the
problem of ice formation. After a rapid glaciation process has been triggered,
wake-induced nucleation might become significant, as the ice concentration
then increases to considerable values. Futhermore, the question of whether or
not wake-induced nucleation alone might trigger such a process has not been
fully resolved yet, since the feedback on the size distribution of ice
particles has been disregarded in this study. In order to assess this
dynamical behavior, further time-resolved information on the (pre-)activation
of ice nuclei in the wake currently appears to be necessary. This requires
microphysical modeling of the nucleation process and knowledge on the spatial
distribution of aerosol particles downstream of the ice particle. A resulting
parametrization may be added to existing cloud models with explicit
microphysics in order to assess the evolution of the size distribution, and
moreover, the relative significance of wake-induced nucleation in comparison
to other secondary ice processes.

#### Data availability.

The datasets are available upon request to the corresponding author.

#### Author contributions.

TL and MU conceptualized the idea. JD provided the simulation code and mesh.
MK conducted the simulations. MK, AC and MU post-processed the data and TL
assisted in the interpretation of the results. MK prepared the manuscript and
AC contributed to it. MU, AC, JD and TL revised the manuscript.

#### Competing interests.

The authors declare that they have no conflict of interest.

#### Acknowledgements

The authors thank Alexei Kiselev for useful discussions. The simulations were
partially performed at the Steinbuch Centre for Computing in Karlsruhe and the
computer resources, technical expertise and assistance provided by this center
are thankfully acknowledged.

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#  Speech Emotion Recognition with Co-Attention based Multi-level Acoustic
Information

###### Abstract

Speech Emotion Recognition (SER) aims to help the machine to understand
human’s subjective emotion from only audio information. However, extracting
and utilizing comprehensive in-depth audio information is still a challenging
task. In this paper, we propose an end-to-end speech emotion recognition
system using multi-level acoustic information with a newly designed co-
attention module. We firstly extract multi-level acoustic information,
including MFCC, spectrogram, and the embedded high-level acoustic information
with CNN, BiLSTM and wav2vec2, respectively. Then these extracted features are
treated as multimodal inputs and fused by the proposed co-attention mechanism.
Experiments are carried on the IEMOCAP dataset, and our model achieves
competitive performance with two different speaker-independent cross-
validation strategies. Our code is available on GitHub.

Index Terms—  Speech emotion recognition, Multimodal fusion, Multi-level
acoustic information, Co-attention mechanism

##  1 Introduction

Automatic recognition of emotions finds several applications such as human-
computer interaction [1] and surveillance [2]. Some researchers propose to
combine acoustic information with textual information and learn high-level
context information to help make the final emotion prediction [3]. However,
the corresponding transcriptions are not always available for most emotion
recognition applications. Besides, the generated text with a current automatic
speech recognition (ASR) system could also introduce word recognition errors
and interfere with the emotion recognition task. Emotion perception from only
audio signals is much easier to implement compared with multimodal emotion
recognition with additional textual and visual signals because single audio
data is easier to be obtained. Transforming the speech emotion recognition
(SER) problem into a multi-level fusion problem by integrating multiple
acoustic information is a potentially effective method to utilize the complete
audio information.

The vast majority of SER problems involve extracting key audio features like
Mel-frequency Cepstral Coefficient (MFCC), Constant-Q Transform (CQT) or
constructing the corresponding spectrogram image to treat the problem as an
image classification problem [4]. Both MFCC and spectrogram reflect more
information of a speech signal in the frequency domain. MFCC can be regarded
as a low-level feature based on human knowledge. Spectrogram can be further
processed to obtain high-level information through a deep neural network.
These methods are intuitive and simple but usually ignore time-domain
information of the speech signal.

Various encoders with different architecture details are designed for
different acoustic signals, e.g., CNN for spectrogram and CNN/LSTM for MFCC.
The acoustic information is mined using a series of CNNs with different kernel
sizes in [3]. Some methods propose to introduce a combination of networks to
extract acoustic information, e.g., [5] combine LSTM and Gated Multi-features
Unit (GMU) to extract both static and dynamic speech signals. In [6], Gao et
al propose a domain-adversarial auto-encoder to extract discriminative
representations with pre-trained spectrogram information. Extracting features
from different sources requires the corresponding source-specific neural
networks.

Different types of attention mechanisms have been proposed for processing the
extracted features, like the commonly used self-attention [5, 7] and cross-
modal attention [8]. For the models with more complex input combinations, new
attention mechanisms are introduced. [9] fuses two modalities and then
combines the result with another modality using the proposed attentive
modality-hop mechanism. In [10], a hierarchical attention-based temporal
convolutional network is designed to fuse the inter-channel and intra-channel
features for spectrogram images.

Fig. 1: The overall architecture of our proposed method.

In this paper, we introduce three different encoders for multiple levels of
acoustic information: CNN for spectrogram, BiLSTM for MFCC and the
transformer-based acoustic extracting network wav2vec2 [11] for raw audio
signals. With the designed co-attention module, we optimize to get the final
wav2vec2 embedding (W2E) after weighting each frame by utilizing the effective
information extracted from MFCC and spectrogram features. We concatenate all
three extracted features and make the final emotion prediction with this
finally fused information. The proposed model surpasses current competitive
models on the widely used IEMOCAP dataset with the leave-one-speaker-out and
leave-one-session-out cross-validation strategy.

##  2 PROPOSED METHOD

In this section, we describe our co-attention-based SER system by integrating
multiple acoustic information. Fig.1 shows the overall structure of our
proposed method. As illustrated, after splitting the raw audio utterance into
several segments, three levels of acoustic information (MFCC, spectrogram and
W2E) of a segment are introduced to the respective feature encoder networks
and fused with the proposed co-attention method for the final emotion
recognition.

###  2.1 Model Overview

We denote the MFCC, spectrogram and wav2vec2, which are obtained from the same
audio segment, as $x_{m}\in\mathbf{R}^{T_{m}\times{D_{m}}}$,
$x_{s}\in\mathbf{R}^{T_{s}\times{D_{s}}}$ and
$x_{w}\in\mathbf{R}^{T_{w}\times{1}}$, respectively. The extracted MFCC
features $x_{m}^{\prime}$ and spectrogram features $x_{s}^{\prime}$ are
concatenated and transformed with linear layers to get the weights for
different frames of wav2vec outputs $x_{w}^{\prime}$. After multiplication
with these generated weights, we get the final W2E vector from the raw wav2vec
outputs. The final obtained W2E $x_{w}^{\prime\prime}$ are concatenated with
the previous MFCC features $x_{m}^{\prime}$ and spectrogram features
$x_{s}^{\prime}$ for the final emotion recognition task. The generated weights
of wav2vec frames from MFCC and spectrogram features and the final feature
combination are denoted as $x_{coatt}^{\prime}$ and $x^{\prime}$,
respectively. The target of the data is denoted by $y$ and the final
prediction is denoted as $\hat{y}$.

###  2.2 Learning with Multi-level Acoustic Information

Here we define the multi-level acoustic information as the combination of the
human knowledge based low-level MFCC, deep learning based high-level
spectrogram and W2E, thus to cover characteristics of the speech signal in
both frequency and time domain. MFCC sequence is processed by a bidirectional
LSTM with a dropout of 0.5 and flattened. The flattened vector is input to a
linear layer with ReLU as an activation function with a dropout of 0.1 to
obtain

$x_{m}^{\prime}=f_{m}(BiLTSM(x_{m}))$ (1)

where $x_{m}^{\prime}\in\mathbf{R}^{D_{m}^{\prime}}$.

The spectrogram image is first reshaped for the pre-trained AlexNet. A similar
operation as for MFCC features is conducted on the AlexNet extracted features
to obtain

$x_{s}^{\prime}=f_{s}(AlexNet(x_{s}))$ (2)

where $x_{s}^{\prime}\in\mathbf{R}^{D_{s}^{\prime}}$.

Raw audio segments are sent directly to the corresponding wav2vec2 processor
and wav2vec2 model to get the target raw wav2vec2 outputs as

$x_{w}^{\prime}=Wav2Vec2(x_{w})$ (3)

where $x_{w}^{\prime}\in\mathbf{R}^{T_{w}^{\prime}\times{D_{w}^{\prime}}}$.

###  2.3 Co-attention-based Fusion

Considering that all three acoustic information sources play a similar role in
the final emotion prediction, we use the correlation among them to guide the
feature adaptation. Generally, the last frame or the average of the wav2vec2
output is used to represent the wav2vec2 features. It is obvious that we lose
some effective information among the sequence dimension. Here, we introduce a
kind of co-attention module to combine different frames of W2E with frame
weights generated by the features of MFCC and spectrogram features.

Firstly, we create a 1-dimension matrix from MFCC features $x_{m}^{\prime}$
and spectrogram features $x_{s}^{\prime}$ with a transformation layer given by

$x_{att}^{\prime}=f_{att}(x_{m}^{\prime}\oplus{x_{s}^{\prime}})$ (4)

where $x_{att}^{\prime}\in\mathbf{R}^{1\times{T_{w}^{\prime}}}$.

The wav2vec2 outputs are multiplied with the previous generated weights to get
the final weighted wav2vec2 features as

$x_{w}^{\prime\prime}=({x_{att}^{\prime}}\cdot x_{w}^{\prime})^{T}$ (5)

where $x_{w}^{\prime\prime}\in\mathbf{R}^{{D_{w}^{\prime}}}$.

The final MFCC, spectrogram features and the weighted W2Es are concatenated
and the speech emotion prediction is written as

$\hat{y}=f(x_{m}^{\prime}\oplus{x_{s}^{\prime}}\oplus{x_{w}^{\prime\prime}})$
(6)

###  2.4 Objective

We use the commonly used cross-entropy loss for emotion classification and our
objective is

$L=L_{ce}(y-\hat{y})$ (7)

Table 1: Performance comparison with 5-fold leave-one-session-out [12,7,13] and 10-fold leave-one-speaker-out [14,15,16,17,18,5] cross-validation strategy on IEMOCAP. Model | WA | UA
---|---|---
CNN-ELM+STC attention[12] | 61.32 | 60.43
Audio25[7] | 60.64$\pm$1.96 | 61.32$\pm$2.26
IS09 - classification [13] | 68.1 | 63.8
Ours | 69.80 | 71.05
RNN(prop.)-ELM[14] | 62.85 | 63.89
3D ACRNN[15] | - | 64.74$\pm$5.44
BLSTM-CTC-CA[16] | 69.0 | 67.0
CNN GRU-SeqCap[17] | 72.73 | 59.71
CNN${\\_}$TF${\\_}$Att.pooling[18] | 71.75 | 68.06
HNSD[5] | 70.5 | 72.5
Ours | 71.64 | 72.70

##  3 EXPERIMENT

Our proposed method is validated on the Interactive Emotional Dyadic Motion
Capture (IEMOCAP) [19] dataset. In this section, we firstly introduce the
dataset processing and audio sources used. Then we describe our experimental
setup and the used validation strategy.

###  3.1 Datasets

IEMOCAP is a widely used emotion recognition dataset, recorded from ten
different actors with audio, video, transcriptions and motion-capture
information. Following others’ work [12, 7, 5], we merge “happy” and “excited”
into the category of “happy” and we consider the 5531 acoustic utterances from
4 emotions, angry, sad, happy and neutral. In order to more accurately
evaluate the performance of the model, we test our model with the 5-fold
leave-one-session-out and the 10-fold leave-one-speaker-out cross-validation
strategy to generate the speaker-independent results. Also, we use the
commonly used weighted accuracy (WA) and the unweighted accuracy (UA) as the
evaluation metrics.

###  3.2 Experimental Setup

The used raw audio signals are sampled at 16 kHz. We spilt each audio
utterance into several segments with a length of 3 seconds. When a segment is
less than 3 seconds, a padding operation with 0 will be applied to this
segment to keep the same length. The final prediction result of an audio
utterance will be decided by all split segments from this utterance.

To make full use of different levels of speech information, we use three kinds
of acoustic information in this SER task, MFCC, spectrogram and W2E. MFCC is a
40-dimension HTK-style Mel frequencies feature that taking into account the
human auditory characteristics. It is extracted from the raw audio segments
with librosa library [20]. Spectrogram and the W2E are the deep features of
audio signals. For spectrogram, a series of 40-ms Hamming windows with a hop
length of 10 ms is applied and here we treat each windowed block as a frame.
Each frame is transformed into a frequency domain with the Discrete Fourier
Transform (DFT) of length 800. The first 200 DFT points are used as input
spectrogram features. We finally get a spectrogram image with a size of
300*200 for each audio segment. Like the multimodal emotion recognition method
[21], W2E are obtained from the pre-trained transformer-based wav2vec2
network. It is the reflection of the deep feature of speech in the time
domain.

This SER system is implemented in PyTorch. The optimizer for the model is
AdamW with a learning rate of 1e-5. The training batch size is 64 and we set
the early stopping setting as 8 epochs. Our code will be available on
Github111https://github.com/Vincent-ZHQ/CA-MSER.

##  4 Results and Analysis

In this section, we present the model performance and design an ablation study
to evaluate the influence of different inputs and used modules. We also
visualize the extracted features of our model with t-distributed stochastic
neighbour embedding (t-SNE) and the final normalized confusion matrix.

Table 2: Ablation study on the proposed model. Model | WA | UA
---|---|---
MFCC | 57.60 | 58.09
Spectrogram | 62.13 | 62.25
W2E | 64.03 | 65.67
MFCC+W2E (w/o co-att) | 64.62 | 65.93
Spectrogram+W2E (w/o co-att) | 66.20 | 67.22
MFCC+Spectrogram+W2E (w/o co-att) | 67.22 | 67.81
W2E (w/ co-att) | 67.55 | 68.65
MFCC+W2E (w/ co-att) | 69.11 | 70.30
Spectrogram+W2E (w/ co-att) | 70.05 | 71.30
MFCC+Spectrogram+W2E (w/ co-att) | 71.64 | 72.70

###  4.1 Results and Comparison

As shown in Table 1, our proposed method could achieve the best performance of
$69.80\%$ and $71.05\%$ in terms of UA and WA for the leave-one-session-out
validation strategy. And for the leave-one-speaker-out validation strategy,
this method could also achieve the highest UA with a value of $72.70\%$. At
the same time, its performance in WA is also competitive with a very similar
result of $71.64\%$ compared with UA on this unbalanced IEMOCAP dataset.

###  4.2 Ablation Study

Our proposed method utilizes the multiple levels of acoustic information,
which contains the time domain and frequency domain. Table 2 shows the
ablation study of model performance with different combinations of acoustic
information. The first three rows are the emotion recognition results with
only one level of acoustic information: MFCC, spectrogram and W2E. W2E
provides better performance than the others for the final emotion recognition.
The next three rows summarize the results from the combination of different
features with W2E. The last four rows present the results of different
combination features with the weighted W2E information after co-attention. The
combination of multiple acoustic information and the proposed co-attention
module are observed to contribute a lot to improve the whole model’s
performance.

The ablation study also shows the effectiveness of the proposed co-attention
mechanism. From the last two rows of Table 2, the co-attention mechanism
further optimizes the fused data and performs better than the direct
concatenating operation with $4.42\%$ and $4.89\%$ improvement on WA and UA,
respectively. As shown in Fig.2, the t-SNE visualization of the weighted W2E
and final combined features after co-attention present a much more clear
classification boundary when compared with the results of the unweighted W2E
and final combined features without co-attention. From Fig. 3, we also observe
that the final classification results of the model with co-attention are much
better than the model without co-attention from the final normalized confusion
matrix.

(a)W2E w/o co-att

(b)W2E w/ co-att

(c)Final features w/o co-att

(d)Final features w/ co-att

Fig. 2: The t-SNE visualization of feature distribution. (a) and (b) are the
final extracted W2Es in the model trained with multi-level acoustic
information without and with the proposed co-attention. (c) and (d) are the
final combined features without and with the proposed co-attention

(a) w/o co-att

(b) w/ co-att

Fig. 3: The normalized confusion matrix for the final speech emotion
recognition without and with the proposed co-attention module.

##  5 Conclusion

This paper proposes a co-attention-based SER system utilizing multi-level
acoustic information. By designing different encoders, this model could get
feature-specific information from the raw audio signals and enables
complementary acoustic information for the SER problem. Also, this method
introduces a co-attention based fusion method for getting weighted wav2vec2
embeddings and combining the final features. The experiments on the IEMOCAP
dataset show that our proposed method achieves competitive performance with
different speaker-independent cross-validation methods. In the future, we
would like to combine the knowledge from different languages or datasets to
improve the final performance.

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