11institutetext: Max-Planck-Institut für Sonnensystemforschung, Justus-von-
Liebig-Weg 3, 37077 Göttingen, Germany 22institutetext: LESIA, Observatoire de
Paris, PSL Research University, CNRS, Université Pierre et Marie Curie,
Université Paris Diderot, 92195 Meudon, France

# Coupling between turbulence and solar-like oscillations: A combined
Lagrangian PDF/SPH approach

II – Mode driving, damping and modal surface effect

J. Philidet Coupling between turbulence and solar-like oscillations: A
combined Lagrangian PDF/SPH approachCoupling between turbulence and solar-like
oscillations: A combined Lagrangian PDF/SPH approachCoupling between
turbulence and solar-like oscillations: A combined Lagrangian PDF/SPH
approachCoupling between turbulence and solar-like oscillations: A combined
Lagrangian PDF/SPH approach K. Belkacem Coupling between turbulence and solar-
like oscillations: A combined Lagrangian PDF/SPH approachCoupling between
turbulence and solar-like oscillations: A combined Lagrangian PDF/SPH approach
M.-J. Goupil Coupling between turbulence and solar-like oscillations: A
combined Lagrangian PDF/SPH approachCoupling between turbulence and solar-like
oscillations: A combined Lagrangian PDF/SPH approach

(Received 18 December 2021 / Accepted 21 April 2022)

###### Abstract

Context. The ever-increasing quality of asteroseismic measurements offers a
unique opportunity to use the observed global acoustic modes to infer the
physical properties of stellar interiors. In solar-like oscillators, the
finite lifetime of the modes allows their amplitudes and linewidths to be
estimated, which provide invaluable information on the highly turbulent
motions at the top of the convective envelope. But exploiting these
observables requires a realistic theoretical framework for the description of
the turbulence–oscillation coupling.

Aims. The first paper of this series established a linear stochastic wave
equation for solar-like $p$-modes, correctly taking the effect of turbulence
thereon into account. In this second paper, we aim at deriving simultaneous
expressions for the excitation rate, damping rate, and modal surface effect
associated with any given $p$-mode, as an explicit function of the statistical
properties of the turbulent velocity field.

Methods. We reduce the stochastic wave equation to complex amplitude equations
for the normal oscillating modes of the system. We then derive the equivalent
Fokker-Planck equation that governs the evolution of the probability density
function jointly associated with the real amplitudes and phases of all the
oscillating modes of the system simultaneously. The effect of the finite-
memory time of the turbulent fluctuations (comparable to the period of the
modes) on the modes themselves is consistently and rigorously accounted for,
by means of the simplified amplitude equation formalism. This formalism
accounts for mutual linear mode coupling in full, and we then turn to the
special single-mode case. This allows us to derive evolution equations for the
mean energy and mean phase of each mode, from which the excitation rate, the
damping rate, and the modal surface effect naturally arise.

Results. The expressions obtained here 1) are written as explicit functions of
the statistical properties of turbulence, thus allowing for any prescription
thereof to be tested against observations, 2) include the contribution of the
turbulent dissipation more realistically, and 3) concern the excitation rate,
the damping rate, and the modal surface effect of the modes simultaneously. We
show that the expression for the excitation rate of the modes is identical to
previous results obtained through a different modelling approach, thus
supporting the validity of the formalism presented here. We also recover the
fact that the damping rate and modal surface effect correspond to the real and
imaginary part of the same single complex quantity. We explicitly separate the
different physical contributions to these observables, in particular the
turbulent pressure contribution and the joint effect of the pressure-rate-of-
strain correlation and the turbulent dissipation. We show that the former
dominates for high-frequency modes and the latter for low-frequency modes. To
illustrate the usefulness of this formalism, we apply it to a simplified case
where we can quantify the relative importance of these two contributions, and
in particular the threshold between the two frequency regimes, as a function
of the turbulent frequency and the degree of anisotropy of both the Reynolds-
stress tensor and the dissipation of turbulent energy.

Conclusions. The formalism developed in these first two papers, applied to the
case of a simplified Lagrangian stochastic model for proof-of-concept
purposes, indeed proves to be viable, relevant, and useful for addressing the
issue of turbulence–oscillation coupling in the context of solar-like
oscillators. It opens the door to subsequent studies physically more
appropriate to the stellar case. It will also allow, once mode coupling is
included (i.e. by going beyond the single-mode case), for a realistic
description of mode-mode scattering and its influence on mode damping, mode
frequency, and the energy distribution across the solar $p$-mode
eigenspectrum.

###### Key Words.:

methods: analytical - Sun: helioseismology - Sun: oscillations

##  1 Introduction

Stellar oscillations are a powerful tool for probing the interior of stars. In
solar-type stars, these oscillations have a finite lifetime, meaning that it
is possible to measure not only the frequency of the resonant modes, but also
their line profile in the Fourier domain. While the frequencies give
invaluable information about the equilibrium structure of the star, the line
profiles carry the signature of the energetic aspects of the modes, which are
tightly related to the physics of the turbulence occurring at the top of the
convective envelope. As a result, solar-like oscillations offer a unique
opportunity to constrain the properties of stellar turbulent convection, which
remains to this day one of the major challenges in stellar physics (e.g. Kupka
& Muthsam 2017).

The coupling between solar-like oscillations and turbulent convection
manifests itself in the seismic observables in several ways. First, it is
responsible for the discrepancy between the observed $p$-mode frequencies and
those theoretically computed from oscillation calculations –- commonly
referred to as the ‘surface effect’ (Dziembowski et al. 1988; Christensen-
Dalsgaard et al. 1996; Rosenthal et al. 1999). This discrepancy is the most
important hurdle on the road to a full exploitation of the asteroseismic
diagnosis potential of solar-like $p$-mode frequencies, and it prevents us
from accurately inferring the internal equilibrium structure of these stars or
their global parameters. It is therefore of utmost importance to correct the
surface effect, a task to which numerous studies have been devoted, using
either theoretical prescriptions (e.g. Gabriel et al. 1975; Balmforth 1992b;
Houdek 1996; Rosenthal et al. 1999; Grigahcène et al. 2005) or empirical
formulations (e.g. Kjeldsen et al. 2008; Christensen-Dalsgaard 2012; Ball &
Gizon 2014; Sonoi et al. 2015).

Secondly, the turbulent convective motions in the superadiabatic region are
responsible for both the stochastic excitation and the linear damping of
solar-like oscillations. As a result, the signature of turbulent convection is
also carried by the observed amplitudes and linewidths of the modes, and the
latter can be used to constrain the former. This task calls for the
development of theoretical prescriptions for both the stochastic driving (e.g.
Goldreich & Keeley 1977a, b; Balmforth 1992a, c; Samadi & Goupil 2001; Chaplin
et al. 2005; Samadi et al. 2005, 2006; Belkacem et al. 2006, 2008, 2010) and
the linear damping (e.g. Goldreich & Kumar 1991; Balmforth 1992a; Grigahcène
et al. 2005; Dupret et al. 2006; Belkacem et al. 2012) of solar-like
$p$-modes.

But because the coupling between turbulent convection and solar-like
oscillations occurs predominantly in the superadiabatic region, where heat
transfers, convective motions, and oscillations all share the same typical
spatial scales and timescales, establishing a theoretical prescription for
this coupling constitutes a challenging task. As a result, simplifying
assumptions are very often necessary to obtain tractable relations. Many
previous theoretical efforts are based on time-dependent mixing-length
theories (e.g. Gabriel et al. 1975; Houdek 1996; Grigahcène et al. 2005; Sonoi
et al. 2017; Houdek et al. 2017, 2019), which amounts to reducing the entire
turbulent flow to a single typical spatial scale and timescale, in direct
contradiction with the picture of the turbulent cascade in high Reynolds
number flows. This constitutes a major flaw of these formalisms, as the
turbulent pressure and the turbulent dissipation both derive directly from
this turbulent cascade and are deemed to play a major role in
turbulence–oscillation coupling. Furthermore, while the mixing-length
hypothesis is relevant in the bulk of the convective envelope, it ceases to be
valid closer to the surface of the star, as highlighted by 3D hydrodynamic
simulations (see Nordlund et al. 2009, for a review). Another problem with
these approaches is the necessity to separate the equations of hydrodynamics
into distinct sets of equations for the turbulent convective motions and the
oscillations, respectively, either in the form of a separation between typical
wavenumbers (e.g. Grigahcène et al. 2005) or by performing horizontal
averaging in a simulation (e.g. Nordlund & Stein 2001). While other approaches
have been investigated as an alternative, such as Reynolds-stress models (e.g.
Xiong et al. 2000) or the direct extraction of modes from 3D large-eddy
simulations (e.g. Belkacem et al. 2019; Zhou et al. 2020), no ideal solution
seems to stand out. Therefore, it remains imperative to seek a more adapted
theoretical framework in which to describe this coupling.

In this series of papers, we investigate a novel modelling approach designed
to address the issue of turbulence–oscillation coupling in the context of
solar-like oscillators, based on Lagrangian stochastic models of turbulence.
In the first paper of this series (Philidet et al. 2021), we used a Lagrangian
stochastic model to obtain a linear stochastic wave equation that is
representative of solar-like $p$-modes and simultaneously contains the effect
of the turbulence on the modes. This linear stochastic wave equation
constitutes our baseline theoretical framework for quantifying
turbulence–oscillation coupling. In this second paper, we exploit it to
directly relate the asteroseismic observables – namely, the amplitudes and
linewidths of the modes, as well as the surface effect incurred by their
eigenfrequencies – to the statistical properties of the underlying turbulence.
The present formalism differs from previous approaches in several ways: 1) the
turbulence is included in its most general form, which allows for any
prescription of turbulent convection to be tested against observations; 2) the
treatment of turbulent dissipation is more realistic than in mixing-length
theory, which allows its impact on turbulence–oscillation coupling to be
quantified; 3) all seismic observables (surface effect, mode amplitudes, and
linewidths) are quantified simultaneously, thus strengthening the constraints
we can put on the properties of turbulent convection.

This paper is structured as follows. In Section 2 we recall the linear
stochastic wave equation we obtained in the first paper of this series
(Philidet et al. 2021), and we give a succinct physical interpretation of the
various terms that it exhibits. In Section 3 we detail how this stochastic
wave equation can be reduced to coupled equations for the complex amplitude of
all the individual normal modes of the system, through the application of the
simplified amplitude equation formalism, and we explicitly derive this
amplitude equation in the case of solar-like $p$-modes. In doing so, we keep
the simplifying assumption that the entropy fluctuations of the gas can be
neglected. In Section 4 we consider the single-mode case, which, for main-
sequence solar-like oscillators, is likely to constitute the dominant
contribution, even in the presence of mode coupling. This yields simultaneous
analytical expressions for the excitation rate, the damping rate, and the
modal surface effect associated with any given mode, as an explicit function
of the statistical properties of the underlying turbulent velocity field in
their most general form, and allows us to separate the contributions to these
quantities. To illustrate the usefulness of the present formalism, we apply it
to a simplified case in Section 4.4, where the relative importance of the
physical contributions to both mode damping and the modal surface effect can
be quantified. Finally, we summarise our findings and draw our conclusions in
Section 5.

##  2 The stochastic wave equation

In this section we summarise the results obtained in Philidet et al. (2021),
hereafter referred to as Paper I. Paper I established a stochastic linear wave
equation for solar-like acoustic modes, which consistently encompass the
effect of the turbulence, and therefore naturally contains the information on
the coupling between turbulence and oscillations. As such, this linear wave
equation is stochastic by nature (i.e. it contains an intrinsically random
part), and the stochastic part represents the effect of the random turbulent
fluctuations on the behaviour of solar-like $p$-modes. To derive the linear
stochastic wave equation, we adopt the three following steps.

First, we consider a Lagrangian stochastic model of turbulence, whereby the
flow is represented by a large set of individual fluid particles. As a first
step, these fluid particles are only characterised by their position and
velocity. This Lagrangian stochastic model is then modified to yield
stochastic differential equations for Eulerian variables, which are more
suited to the description of oscillations. This operation leads us to two
stochastic differential equations governing the evolution of the fluid
displacement $\bm{\xi}(\mathbf{x},t)$ and velocity $\mathbf{u}(\mathbf{x},t)$
(see Eqs. 25 and 26 of Paper I).

Second, these two equations rely on the estimate of the instantaneous ensemble
average of several fluid quantities – namely the density, pressure, velocity
and Reynolds-stress tensor. These instantaneous ensemble averages are
estimated directly from the set of fluid particles, in a similar fashion to
the smoothed particle hydrodynamics (SPH) formalism. In particular, this step
requires the introduction of a kernel function $K(\mathbf{r})$, which serves
as a spatial weighting function designed to filter the neighbouring fluid
particles for the estimation of the local instantaneous ensemble averages.
This step leads us to explicit expressions for these averages, as a function
of $\bm{\xi}(\mathbf{x},t)$ and $\mathbf{u}(\mathbf{x},t)$ only (see Eqs. 32
to 35 in Paper I).

Third, we then treat the turbulent part of the displacement variable
$\bm{\xi}_{\text{t}}$ (resp. $\mathbf{u}_{\text{t}}$ for the velocity
variable) as an external input, while the residual
$\bm{\xi}_{\text{osc}}\equiv\bm{\xi}-\bm{\xi}_{\text{t}}$ (resp.
$\mathbf{u}_{\text{osc}}\equiv\mathbf{u}-\mathbf{u}_{\text{t}}$) represent the
oscillatory component of these variables, including their coupling with
turbulence. We note that the equations themselves are not split into
turbulence equations and oscillation equations, so that we retain one system
of equations for both at the same time, instead of two separate sets. The full
system of equations – namely the fluid equations obtained in step 1) and the
mean field expressions obtained in step 2) – is then linearised in terms of
the two wave variables $\bm{\xi}_{\text{osc}}$ and $\mathbf{u}_{\text{osc}}$.
This is done by adopting a certain number of hypotheses and approximations,
all of which are explicitly itemised in Paper I (see hypotheses H1 to H5 in
Section 3.1 therein, as well as H6 to H8 in Section 4 therein). Of particular
importance is the adiabatic approximation, which we adopted as a first step,
and whereby we considered any fluid parcel to conserve its entropy during its
evolution. This freed us from having to include an entropy or energy equation
in the system, and we instead adopted a polytropic relation between gas
pressure and density. In the following, this is what we mean by ‘adiabatic’.
By contrast, we do not consider the oscillations to be adiabatic in the
mechanical sense of the term, meaning that they can still exchange energy with
the background, thus allowing the contributions of both the driving and the
damping of the modes to be included (see discussion in Section 4).

Ultimately, this allows us to establish the following linear stochastic wave
equation for solar-like acoustic modes in the presence of turbulent convection
(see Eqs. 42 and 43 in Paper I):

$\displaystyle\dfrac{\partial\bm{\xi}_{\mathrm{osc}}}{\partial
t}-\mathbf{u}_{\mathrm{osc}}-(\bm{\xi}_{\mathrm{osc}}\cdot\bm{\nabla})\mathbf{u}_{\text{t}}-(\bm{\xi_{t}}\cdot\bm{\nabla})\mathbf{u}_{\mathrm{osc}}=(\bm{\xi_{t}}\cdot\bm{\nabla})\mathbf{u}_{\text{t}}\leavevmode\nobreak\
,$ (1) $\displaystyle\dfrac{\partial\mathbf{u}_{\mathrm{osc}}}{\partial
t}-\mathbf{L}_{1}^{d}-\mathbf{L}_{1}^{s}=\mathbf{L}_{0}\leavevmode\nobreak\ .$
(2)

The vectors $\mathbf{L}_{0}$, $\mathbf{L}_{1}^{d}$, and $\mathbf{L}_{1}^{s}$
are given by (see Eqs. 44 to 46 in Paper I)

$L_{1,i}^{d}=\dfrac{1}{\rho_{0}}\left(\dfrac{1}{\rho_{0}}\partial_{i}p_{0}-\partial_{i}c_{0}^{2}\right)\mathfrak{I}\left(\xi_{\mathrm{osc,j}}\partial_{j}K^{\mathbf{x}}\right)+\dfrac{c_{0}^{2}}{\rho_{0}}\mathfrak{I}\left(\xi_{\mathrm{osc,j}}\partial_{j}\partial_{i}K^{\mathbf{x}}\right)\\\
+G_{ij,0}\left(u_{\mathrm{osc,j}}-\dfrac{1}{\rho_{0}}\mathfrak{I}\left(u_{\mathrm{osc,j}}K^{\mathbf{x}}\right)\right)\leavevmode\nobreak\
,$ (3)
$L_{1,i}^{s}=-u_{\mathrm{osc,j}}\partial_{j}u_{\text{t},i}-u_{\text{t},j}\partial_{j}u_{\mathrm{osc,i}}-G_{ij,0}\dfrac{1}{\rho_{0}}\mathfrak{I}\left(\xi_{\mathrm{osc,k}}\partial_{k}\left(u_{\text{t},j}K^{\mathbf{x}}\right)\right)\\\
+\left(\dfrac{\partial
G_{ij}}{\partial\widetilde{u_{k}^{\prime\prime}u_{l}^{\prime\prime}}}\widetilde{u_{k}^{\prime\prime}u_{l}^{\prime\prime}}_{(1)}+\dfrac{\partial
G_{ij}}{\partial(\partial_{k}\widetilde{u_{l}})}\partial_{k}\widetilde{u_{l}}_{(1)}+\dfrac{1}{2}\dfrac{\partial
G_{ij}}{\partial\epsilon}\omega_{t}\widetilde{u_{i}^{\prime\prime}u_{i}^{\prime\prime}}_{(1)}\right)u_{\text{t},j}\\\
+\dfrac{1}{4}\sqrt{\dfrac{2C_{0}\omega_{t}}{\widetilde{u_{i}^{\prime\prime}u_{i}^{\prime\prime}}_{0}}}\widetilde{u_{i}^{\prime\prime}u_{i}^{\prime\prime}}_{(1)}\eta_{i}$
(4)

and

$L_{0,i}=-\dfrac{1}{\rho_{0}}\partial_{j}\left(\vphantom{\dfrac{1}{2}}\rho_{0}u_{\text{t},i}u_{\text{t},j}-\rho_{0}\overline{u_{\text{t},i}u_{\text{t},j}}\right)\leavevmode\nobreak\
,$ (5)

where we have defined

$\mathfrak{I}(f)\equiv\displaystyle\int\mathop{}\\!\mathrm{d}^{3}\mathbf{y}\leavevmode\nobreak\
\rho_{0}(\mathbf{y})f(\mathbf{y})\leavevmode\nobreak\ .$ (6)

The indices ‘0’ and ‘1’ refer to equilibrium values and fluctuations around
the equilibrium values, respectively. The notation $\partial_{i}$ denotes the
$i$-th component of the gradient operator, and we have used Einstein summation
convention. The dimensionless quantity $C_{0}$ denotes the Kolmogorov
constant, for which a commonly accepted experimental value is $C_{0}=2.1$
(e.g. Haworth & Pope 1986). We have introduced the centred kernel function
$K^{\mathbf{x}}(\mathbf{y})\equiv K(\mathbf{y}-\mathbf{x})$, and the
perturbations of the Reynolds-stress tensor and mean shear tensor are given by
(see Eqs. 47 and 48 in Paper I)

$\rho_{0}\widetilde{u_{i}^{\prime\prime}u_{j}^{\prime\prime}}_{(1)}=-\widetilde{u_{i}^{\prime\prime}u_{j}^{\prime\prime}}_{0}\mathfrak{I}\left(\xi_{\mathrm{osc,k}}\partial_{k}K^{\mathbf{x}}\right)\\\
+\mathfrak{I}\left(\xi_{\mathrm{osc,k}}\partial_{k}\left(u_{\text{t},i}u_{\text{t},j}K^{\mathbf{x}}\right)+u_{\text{t},i}u_{\mathrm{osc,j}}K^{\mathbf{x}}+u_{\text{t},j}u_{\mathrm{osc,i}}K^{\mathbf{x}}\right)$
(7)

and

$\rho_{0}(\partial_{i}\widetilde{u_{j}})_{(1)}=-\mathfrak{I}\left(u_{\mathrm{osc,j}}\partial_{i}K^{\mathbf{x}}+\xi_{\mathrm{osc,k}}\partial_{k}\left(u_{\text{t},j}\partial_{i}K^{\mathbf{x}}\right)\right)\\\
-\dfrac{1}{\rho_{0}}\partial_{i}\rho_{0}\leavevmode\nobreak\
\mathfrak{I}\left(u_{\mathrm{osc,j}}K^{\mathbf{x}}+\xi_{\mathrm{osc,k}}\partial_{k}\left(u_{\text{t},j}K^{\mathbf{x}}\right)\right)\leavevmode\nobreak\
.$ (8)

The quantities $\rho_{0}$, $p_{0}$, and $\gamma$ are the equilibrium density,
equilibrium gas pressure, and polytropic exponent, respectively, and
$c_{0}^{2}\equiv p_{0}\gamma/\rho_{0}$ is the equilibrium sound speed squared.
Finally, by construction, $\eta_{i}(\mathbf{x},t)$ is defined as a
multivariate Gaussian process whose values at two distinct locations are
completely uncorrelated, and which verifies

$\displaystyle\overline{\bm{\eta}}=\mathbf{0}\leavevmode\nobreak\ ,$ (9)
$\displaystyle\overline{\eta_{i}(\mathbf{x},t)\eta_{j}(\mathbf{x},t^{\prime})}=\delta(t-t^{\prime})\delta_{ij}\leavevmode\nobreak\
,$ (10)

where $\delta_{ij}$ is the Kronecker symbol, $\delta$ is the Dirac
distribution, and $\overline{\vphantom{a^{a}}\leavevmode\nobreak\
.\leavevmode\nobreak\ }$ denotes the ensemble average.

Equations 1 and 2 put together govern the time evolution of the wave variables
$\bm{\xi}_{\mathrm{osc}}$ and $\mathbf{u}_{\mathrm{osc}}$ as an explicit
function of either the wave variables themselves or the turbulent fields
$\bm{\xi}_{\text{t}}$ and $\mathbf{u}_{\text{t}}$. More specifically, Eq. 2 is
split three ways. $\mathbf{L}_{1}^{d}$ contains all the terms that are linear
in the wave variables, but independent of the turbulent fields (they are
therefore deterministic). $\mathbf{L}_{1}^{s}$ contains all the terms that are
linear in the wave variables, and also depend on the turbulent fields (they
are therefore stochastic). Finally, $\mathbf{L}_{0}$ contains all the terms
that are independent of the wave variables, and encompass all the
inhomogeneous forcing terms.

Formally, the oscillations can be described in a Hilbert space of infinite
dimension, where the ket $\ket{\bm{z}}$, in Dirac notation, is defined as111We
note that the oscillatory displacement, $\bm{\xi}_{\mathrm{osc}}$, is
multiplied by the angular frequency, $\omega$, of the mode under consideration
such that all components of $\ket{\bm{z}}$ have the same dimension, i.e. that
of a velocity. This frequency must not be confused with the turbulent
frequency $\omega_{t}$ introduced in the wave equation (see Paper I for more
details).

$\ket{\bm{z}}\equiv\Ket{\left\\{\omega\bm{\xi}_{\mathrm{osc}}(\mathbf{x})\leavevmode\nobreak\
;\leavevmode\nobreak\
\mathbf{u}_{\mathrm{osc}}(\mathbf{x})\right\\}_{\mathbf{x}\in\mathcal{V}}}\leavevmode\nobreak\
,$ (11)

where $\mathcal{V}$ is the volume of the star, and $\omega$ refers to the
angular frequency of the mode under consideration. Then, Eqs. 1 and 2 can be
rewritten in the following way:

$\dfrac{\mathop{}\\!\mathrm{d}\Ket{\bm{z}}}{\mathop{}\\!\mathrm{d}t}=\mathcal{L}^{d}\ket{\bm{z}}+\mathcal{L}^{s}(t)\ket{\bm{z}}+\ket{\Theta(t)}\leavevmode\nobreak\
,$ (12)

where the deterministic, time-independent linear operator $\mathcal{L}^{d}$ is
given by

$\Ket{\mathcal{L}^{d}|\bm{z}}=\Ket{\left\\{\omega\mathbf{u}_{\mathrm{osc}}\leavevmode\nobreak\
;\leavevmode\nobreak\
\mathbf{L}_{1}^{d}\right\\}_{\mathbf{x}\in\mathcal{V}}}\leavevmode\nobreak\ ,$
(13)

the stochastic, time-dependent linear operator $\mathcal{L}^{s}(t)$ is given
by

$\Ket{\mathcal{L}^{s}(t)|\bm{z}}=\Ket{\left\\{\omega(\bm{\xi}_{\mathrm{osc}}\cdot\bm{\nabla})\mathbf{u}_{\text{t}}+\omega(\bm{\xi}_{\text{t}}\cdot\bm{\nabla})\mathbf{u}_{\mathrm{osc}}\leavevmode\nobreak\
;\leavevmode\nobreak\
\mathbf{L}_{1}^{s}\right\\}_{\mathbf{x}\in\mathcal{V}}}\leavevmode\nobreak\ ,$
(14)

the stochastic vector $\Ket{\Theta(t)}$ is given by

$\Ket{\Theta(t)}=\Ket{\left\\{\omega(\bm{\xi}_{\text{t}}\cdot\bm{\nabla})\mathbf{u}_{\text{t}}\leavevmode\nobreak\
;\leavevmode\nobreak\
\mathbf{L}_{0}\right\\}_{\mathbf{x}\in\mathcal{V}}}\leavevmode\nobreak\ ,$
(15)

and we recall that the vectors $\mathbf{L}_{1}^{d}$, $\mathbf{L}_{1}^{s}$, and
$\mathbf{L}_{0}$ are given respectively by Eqs. 3, 4 and 5. We note that,
since the space dependence of the wave variables is already contained in the
infinite dimension of the Hilbert space where the oscillations are described,
$\ket{\bm{z}}$ only depends on time, such that
$\mathop{}\\!\mathrm{d}\leavevmode\nobreak\ /\leavevmode\nobreak\
\mathop{}\\!\mathrm{d}t$ is a total time derivative.

The impact of turbulent convection on the frequency and the damping rate of
the acoustic modes is contained within the linear stochastic operator
$\mathcal{L}^{s}(t)$, whereas the driving of the acoustic modes is contained
within the additive noise vector $\ket{\Theta(t)}$. Because of their
stochastic nature, a large portion of these two quantities is actually
incoherent or statistically uncorrelated with the acoustic modes themselves,
and therefore incapable of efficiently impacting the oscillations. It is
therefore necessary to filter in the part of these stochastic perturbations
that does lead to a significant impact on the modes. A naive way to do this
would be to simply take the ensemble average of Eq. 12 (or, equivalently, of
Eqs. 1 and 2). However, this would not work, because even the zero-mean
stochastic part of the turbulent perturbations, being auto-correlated over
timescales comparable to the period of the modes, can impact the behaviour of
the oscillations. It is therefore necessary to incorporate the effective
impact of the finite memory time of the stochastic perturbations on the
oscillations. This can be done in the framework of the simplified amplitude
equation formalism (Stratonovich 1965; Buchler & Goupil 1984; Buchler et al.
1993).

##  3 The simplified amplitude equations

The goal of this section is to use the simplified amplitude equation formalism
to properly account for the effect of convective noise on the excitation and
damping rates of the normal modes of oscillation. To do so, we proceeded in a
two-step approach.

First, the linear stochastic wave equation is transformed into a finite set of
stochastic differential equations governing the evolution of the complex
amplitudes of the normal modes of oscillation. This is done by treating the
stochastic part of the wave equation as perturbations to an otherwise
deterministic wave equation. This leads to a set of coupled amplitude
equations (Buchler & Goupil 1984). We present this first step in Section 3.1.

The amplitude equations contain a stochastic part that, as mentioned above,
has a finite memory time. In a second step, these are substituted for
simplified amplitude equations, which are also stochastic differential
equations governing the evolution of the complex amplitude of the modes, but
whose stochastic part, unlike the original amplitude equations, has zero
memory time, and therefore has no long-term impact on the oscillations. This
step involves two procedures. First the original stochastic amplitude
equations are substituted for an equation for the probability density function
(PDF) of the complex amplitude of the modes, in the form of a Fokker-Planck
equation. We present this Fokker-Planck equation in Section 3.2. Then another
set of stochastic differential equations is constructed, under the constraints
that 1) it must reduce to the exact same Fokker-Planck equation, while 2) only
involving Markov processes (i.e. stochastic processes with zero memory time).
We derive these simplified amplitude equations in Section 3.3.

###  3.1 The full amplitude equations

The stochastic amplitude equations are obtained in two steps (Stratonovich
1965): first deterministic amplitude equations are derived as if there were no
stochastic perturbations; then the stochastic part is treated in a
perturbative framework, and stochastic corrections are added to the
deterministic amplitude equations after the fact. The non-perturbed wave
equation is obtained by setting both $\mathcal{L}^{s}(t)$ and
$\Ket{\Theta(t)}$ to zero in Eq. 12, which yields

$\dfrac{\mathop{}\\!\mathrm{d}\Ket{\bm{z}}}{\mathop{}\\!\mathrm{d}t}=\mathcal{L}^{d}\ket{\bm{z}}\leavevmode\nobreak\
.$ (16)

The linear operator $\mathcal{L}^{d}$ can be diagonalised, and any vector
$\ket{\bm{z}}$ can be decomposed on a basis of eigenvectors for
$\mathcal{L}^{d}$. Because this linear operator is real, the associated
eigenvalues are either real or come in pairs of complex conjugates. Each such
pair of eigenvectors with complex-conjugated eigenvalues correspond to a
single mode of oscillation of the system; we denote them as $\ket{\Psi^{\mu}}$
and $\ket{\Psi^{\dagger\mu}}$ (where the exponent notation $\mu$ is used to
index the different modes) and their associated eigenvalues as
$\kappa_{\mu}\pm\mathrm{j}\omega_{\mu}$, where $\mathrm{j}$ denotes the
imaginary unit.

Without loss of generality, we consider that the eigenvectors are normalised
to unity,

$\Braket{\Psi^{\mu}}{\Psi^{\mu}}=\Braket{\Psi^{\dagger\mu}}{\Psi^{\dagger\mu}}=1\leavevmode\nobreak\
,$ (17)

where $\braket{.}{.}$ refers to the scalar product in the $\ket{\bm{z}}$
Hilbert space. This scalar product needs to be specified. A necessary
condition is that distinct eigenvectors $\ket{\Psi^{\mu}}$ of the non-
perturbed wave equation (Eq. 16) – that is, eigenvectors of the linear
operator $\mathcal{L}^{d}$ – must be orthogonal to one another in the sense of
this scalar product. For adiabatic, non-radial oscillations, this is verified
by the following scalar product (Unno et al. 1989):

$\braket{\psi}{\phi}\equiv\displaystyle\int\mathop{}\\!\mathrm{d}^{3}\mathbf{x}\leavevmode\nobreak\
\rho_{0}(\mathbf{x})\left(\vphantom{f_{1}^{\tau}}\bm{\psi}_{\xi}(\mathbf{x})\cdot\bm{\phi}_{\xi}^{\star}(\mathbf{x})+\bm{\psi}_{u}(\mathbf{x})\cdot\bm{\phi}_{u}^{\star}(\mathbf{x})\right)\leavevmode\nobreak\
,$ (18)

where $\bm{\psi}_{\xi}(\mathbf{x})$ (resp. $\bm{\psi}_{u}(\mathbf{x})$) is the
part of $\ket{\psi}$ associated with the oscillatory displacement (resp.
velocity) at location $\mathbf{x}$. For any eigenvector $\ket{\Psi}$ of the
problem, these two quantities are simply the normalised displacement and
velocity eigenfunctions. Because of the normalisation condition (Eq. 17), they
are related to the non-normalised eigenfunctions $\bm{\xi}_{\mathrm{osc}}$ and
$\mathbf{u}_{\mathrm{osc}}$ through (see Section B.4 for more details)

$\bm{\Psi}^{\mu}_{u}=\mathrm{j}\bm{\Psi}^{\mu}_{\xi}=\dfrac{\mathbf{u}_{\mathrm{osc}}^{\mu}}{\sqrt{2\mathcal{I}\omega_{\mu}^{2}}}=\mathrm{j}\dfrac{\bm{\xi}_{\mathrm{osc}}^{\mu}}{\sqrt{2\mathcal{I}_{\mu}}}\leavevmode\nobreak\
,$ (19)

where $\mathcal{I}_{\mu}$ is the inertia of the mode, defined by

$\mathcal{I}_{\mu}\equiv\displaystyle\int\mathop{}\\!\mathrm{d}^{3}\mathbf{x}\leavevmode\nobreak\
\rho_{0}(\mathbf{x})\left|\bm{\xi}_{\mathrm{osc}}^{\mu}(\mathbf{x})\right|^{2}\leavevmode\nobreak\
.$ (20)

The general solution of Eq. 16 is (Buchler et al. 1993)

$\Ket{\bm{z}(t)}={{\sum}}_{\mu}\dfrac{1}{2}a_{\mu}(t)\exp^{\mathrm{j}\omega_{\mu}t}\Ket{\Psi^{\mu}}+\text{c.c}\leavevmode\nobreak\
,$ (21)

where ‘c.c’ denotes the complex conjugate. In general, $\ket{\bm{z}}$ would be
written as an arbitrary linear combination of $\ket{\Psi^{\mu}}$ and
$\ket{\Psi^{\dagger\mu}}$; however, if $\ket{\bm{z}}$ is initially real, Eq.
16 shows that it will remain real at all later times (because
$\mathcal{L}^{d}$ itself is real), so the $\ket{\Psi^{\dagger\mu}}$ component
of $\ket{\bm{z}}$ is necessarily the complex conjugate of its
$\ket{\Psi^{\mu}}$ component. As for $a_{\mu}(t)$, it denotes the slowly
varying complex amplitude of the mode $\mu$. In the general case, plugging the
solution given by Eq. 21 into Eq. 16, one finds the following, very simple
amplitude equation for $a_{\mu}(t)$:

$\dfrac{\mathop{}\\!\mathrm{d}a_{\mu}}{\mathop{}\\!\mathrm{d}t}=\kappa_{\mu}a_{\mu}\leavevmode\nobreak\
.$ (22)

As expected, while the imaginary part $\omega_{\mu}$ of the eigenvalue
associated with the mode represents its oscillatory angular frequency, its
real part $\kappa_{\mu}$ represents the rate at which its amplitude varies –
either a growing rate or a damping rate, depending on the intrinsic stability
of the mode in the absence of turbulence. However, in the present adiabatic
case, since the operator $\mathcal{L}^{d}$ is Hermitian, we actually have
$\kappa_{\mu}=0$. The quantity $\kappa_{\mu}$ therefore contains the
contribution of all non-adiabatic effects to the damping of the modes. As it
contributes to defining the variation timescale of the mode amplitudes, we
keep this term; however, we do not consider it more explicitly, as our present
goal is to estimate the additional contribution coming from the interaction
with turbulent convection.

Turning now to the full, stochastically perturbed wave equation (Eq. 12), the
addition of the two terms $\mathcal{L}^{s}(t)\ket{\bm{z}}$ and
$\ket{\Theta(t)}$ leads to the following modifications to the amplitude
equations (Buchler et al. 1993):

$\dfrac{\mathop{}\\!\mathrm{d}a_{\mu}}{\mathop{}\\!\mathrm{d}t}=\kappa_{\mu}a_{\mu}+c_{1\mu}\leavevmode\nobreak\
\exp^{-\mathrm{j}\omega_{\mu}t}+{{\sum}}_{\nu}c_{2\mu\nu}\leavevmode\nobreak\
a_{\nu}\leavevmode\nobreak\ \exp^{-\mathrm{j}(\omega_{\mu}-\omega_{\nu})t}\\\
+{{\sum}}_{\nu}c_{3\mu\nu}\leavevmode\nobreak\
a_{\nu}^{\star}\exp^{-\mathrm{j}(\omega_{\mu}+\omega_{\nu})t}\leavevmode\nobreak\
,$ (23)

where

$c_{1\mu}\equiv 2\Braket{\Psi^{\mu}}{\Theta}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\
c_{2\mu\nu}\equiv\Braket{\Psi^{\mu}}{\mathcal{L}^{s}}{\Psi^{\nu}}\leavevmode\nobreak\
\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ \leavevmode\nobreak\
c_{3\mu\nu}\equiv\braket{\Psi^{\mu}}{\mathcal{L}^{s}}{\Psi^{\dagger\nu}}\leavevmode\nobreak\
.$ (24)

The form of the stochastic linear operator $\mathcal{L}_{s}(t)$ and stochastic
vector $\Ket{\Theta(t)}$ being completely specified by Eqs. 14 and 15,
respectively, and the scalar product being defined by Eq. 18, the stochastic
processes $c_{i}(t)$ ($i=1,2,3$) constitute ‘knowns’ of the system. These are
given by

$\displaystyle c_{1\mu}=2\leavevmode\nobreak\
\mathfrak{I}\left(\vphantom{f_{1}^{\tau}}\omega_{\mu}\Psi_{\xi,i}^{\mu}\xi_{\text{t},j}\partial_{j}u_{\text{t},i}+\Psi_{u,i}^{\mu}L_{0,i}\right)$
(25) $\displaystyle
c_{3\mu\nu}=\mathfrak{I}\left(\vphantom{f_{1}^{\tau}}\Psi_{\xi,i}^{\mu}\Psi_{\xi,j}^{\nu}\partial_{j}u_{\text{t},i}+\omega_{\mu}\Psi_{\xi,i}^{\mu}\xi_{\text{t},j}\partial_{j}\Psi_{u,i}^{\nu}+\Psi_{u,i}^{\mu}L_{1,i}^{s,\nu}\right)\leavevmode\nobreak\
,$ (26)

where $L_{1,i}^{s,\nu}$ is obtained by replacing $\bm{\xi}_{\mathrm{osc}}$ and
$\mathbf{u}_{\mathrm{osc}}$ by the eigenfunctions $\bm{\Psi}_{\xi}^{\nu}$ and
$\bm{\Psi}_{u}^{\nu}$ in Eq. 4.

It will be more practical, in the following, to separate the complex amplitude
of the modes into their amplitude $A_{\mu}(t)$ and phase $\Phi_{\mu}(t)$, both
being real functions of time, such that

$a_{\mu}(t)=A_{\mu}(t)\exp^{\mathrm{j}\Phi_{\mu}(t)}\leavevmode\nobreak\ .$
(27)

In place of a single complex stochastic equation for the evolution of
$a_{\mu}(t)$, we obtain two real stochastic equations for the evolution of
$A_{\mu}(t)$ and $\Phi_{\mu}(t)$, which take the following form:

$\displaystyle\dfrac{\mathop{}\\!\mathrm{d}A_{\mu}}{\mathop{}\\!\mathrm{d}t}\equiv
G_{\mu}(A_{\nu},\Phi_{\nu},t)$
$\displaystyle=\kappa_{\mu}A_{\mu}+\mathrm{Re}\left[c_{1\mu}\exp^{-\mathrm{j}(\omega_{\mu}t+\Phi_{\mu})}+{{\sum}}_{\nu}A_{\nu}c_{2\mu\nu}\exp^{-\mathrm{j}(\omega_{\mu}-\omega_{\nu})t-\mathrm{j}(\Phi_{\mu}-\Phi_{\nu})}\right.$
$\displaystyle\hskip
14.22636pt\left.+{{\sum}}_{\nu}A_{\nu}c_{3\mu\nu}\exp^{-\mathrm{j}(\omega_{\mu}+\omega_{\nu})t-\mathrm{j}(\Phi_{\mu}+\Phi_{\nu})}\right]\leavevmode\nobreak\
,$ (28)

and

$\displaystyle\dfrac{\mathop{}\\!\mathrm{d}\Phi_{\mu}}{\mathop{}\\!\mathrm{d}t}\equiv
H_{\mu}(A_{\nu},\Phi_{\nu},t)$
$\displaystyle=\dfrac{1}{A_{\mu}}\mathrm{Im}\left[c_{1\mu}\exp^{-\mathrm{j}(\omega_{\mu}t+\Phi_{\mu})}+{{\sum}}_{\nu}A_{\nu}c_{2\mu\nu}\exp^{-\mathrm{j}(\omega_{\mu}-\omega_{\nu})t-\mathrm{j}(\Phi_{\mu}-\Phi_{\nu})}\right.$
$\displaystyle\hskip
14.22636pt\left.+{{\sum}}_{\nu}A_{\nu}c_{3\mu\nu}\exp^{-\mathrm{j}(\omega_{\mu}+\omega_{\nu})t-\mathrm{j}(\Phi_{\mu}+\Phi_{\nu})}\right]\leavevmode\nobreak\
,$ (29)

where $\mathrm{Re}$ and $\mathrm{Im}$ denote the real and imaginary parts,
respectively. We warn the reader that the function
$G_{\mu}(A_{\nu},\Phi_{\nu},t)$ defined by the right-hand side of Eq. 28 must
not be confused with the drift tensor $G_{ij}$ appearing in Eqs. 3 and 4.

###  3.2 The Fokker-Planck equation for mode amplitude

A fundamental hypothesis we need to make concerning $\mathcal{L}^{s}(t)$ and
$\ket{\Theta(t)}$ is that their correlation timescale is very small compared
to the timescale over which the amplitude and the phase of the mode typically
vary. In the context of solar-like oscillations, the correlation timescale of
these stochastic perturbations correspond to the turnover time of the
turbulent eddies, which, close to the surface of the star, is similar to the
period of the $p$-modes (i.e. $\sim 5$ minutes). On the other hand, the
amplitude and phase of the modes vary over typical timescales that correspond
to their life time, which is indeed much longer than their period ($\tau\sim
3$ hours for the shortest-lived solar modes). As such, this hypothesis is
largely verified in solar-like oscillators. Consequently, all the turbulent
perturbations in the wave equation (Eq. 12) can be approximated by Markov
processes (in the sense that their memory time, while finite, is much smaller
than the evolution timescale of the amplitude of the modes), and subsequently,
so can the stochastic processes $A_{\mu}(t)$ and $\Phi_{\mu}(t)$. It is then
possible to replace the set of $2N$ (where $N$ is the total number of modes)
stochastic differential equations on these quantities (i.e. Eqs. 28 and 29)
with an equivalent, single Fokker-Planck equation governing the evolution of
their joint PDF $w(A_{\mu},\Phi_{\mu},t)$, but whose coefficients are
carefully defined to incorporate in a rigorous manner the effect of the finite
memory time of the processes $A_{\mu}(t)$ and $\Phi_{\mu}(t)$. The Fokker-
Planck equation takes the general form (Stratonovich 1965)

$\dfrac{\partial w}{\partial t}=-\dfrac{\partial w\mathcal{G}_{\mu}}{\partial
A_{\mu}}-\dfrac{\partial
w\mathcal{H}_{\mu}}{\partial\Phi_{\mu}}+\dfrac{1}{2}\dfrac{\partial^{2}w\mathcal{D}_{\mu\nu}}{\partial
A_{\mu}\partial
A_{\nu}}+\dfrac{1}{2}\dfrac{\partial^{2}w\mathcal{E}_{\mu\nu}}{\partial
A_{\mu}\partial\Phi_{\nu}}+\dfrac{1}{2}\dfrac{\partial^{2}w\mathcal{F}_{\mu\nu}}{\partial\Phi_{\mu}\partial\Phi_{\nu}}\leavevmode\nobreak\
,$ (30)

where $\mathcal{G}_{\mu}$ and $\mathcal{H}_{\mu}$ represent the probability
fluxes in ($A_{\mu},\Phi_{\mu}$) space, and $\mathcal{D}_{\mu\nu}$,
$\mathcal{E}_{\mu\nu}$ and $\mathcal{F}_{\mu\nu}$ are the elements of the
$2N\times 2N$ probability diffusion matrix. The explicit computation of the
probability fluxes and diffusion coefficients is detailed in Appendix A, and
these coefficients read

$\mathcal{G}_{\mu}=A_{\mu}\left[\kappa_{\mu}+{{\sum}}_{\nu\neq\mu}\mathrm{Re}\left(\alpha_{2\mu\nu}^{(a)}\right)+\alpha_{2\mu}^{(R)}+{{\sum}}_{\nu}\mathrm{Re}\left(\alpha_{3\mu\nu}^{(a)}\right)\right]\\\
+\dfrac{1}{2A_{\mu}}\left[\mathrm{Re}\left(\alpha_{1\mu}\right)+{{\sum}}_{\nu\neq\mu}A_{\nu}^{2}\mathrm{Re}\left(\alpha_{2\mu\nu}^{(b)}\right)+{{\sum}}_{\nu}A_{\nu}^{2}\mathrm{Re}\left(\alpha_{3\mu\nu}^{(b)}\right)\right]\leavevmode\nobreak\
,$ (31)
$\mathcal{H}_{\mu}={{\sum}}_{\nu\neq\mu}\mathrm{Im}\left(\alpha_{2\mu\nu}^{(a)}\right)+{{\sum}}_{\nu}\mathrm{Im}\left(\alpha_{3\mu\nu}^{(a)}\right)\leavevmode\nobreak\
,$ (32)
$\mathcal{D}_{\mu\nu}=\delta_{\mu\nu}\left[\mathrm{Re}\left(\alpha_{1\mu}\right)+{{\sum}}_{\lambda\neq\mu}A_{\lambda}^{2}\mathrm{Re}\left(\alpha_{2\mu\lambda}^{(b)}\right)\right.\\\
\left.+2A_{\mu}^{2}\alpha_{2\mu}^{(R)}+{{\sum}}_{\lambda}A_{\lambda}^{2}\mathrm{Re}\left(\alpha_{3\mu\lambda}^{(b)}\right)\right]\\\
+\dfrac{1}{2}(1-\delta_{\mu\nu})A_{\mu}A_{\nu}\left[\mathrm{Re}\left(\alpha_{2\mu\nu}^{(a)}\right)+\mathrm{Re}\left(\alpha_{3\mu\nu}^{(a)}\right)+\mathrm{sym.}\right]\leavevmode\nobreak\
,$ (33) $\mathcal{E}_{\mu\nu}=0\leavevmode\nobreak\ ,$ (34)
$\mathcal{F}_{\mu\nu}=\delta_{\mu\nu}\left[\dfrac{1}{A_{\mu}^{2}}\mathrm{Re}\left(\alpha_{1\mu}\right)+\dfrac{1}{A_{\mu}^{2}}{{\sum}}_{\lambda\neq\mu}A_{\lambda}^{2}\mathrm{Re}\left(\alpha_{2\mu\lambda}^{(b)}\right)\right.\\\
\left.+2\alpha_{2\mu}^{(I)}+\dfrac{1}{A_{\mu}^{2}}{{\sum}}_{\lambda}A_{\lambda}^{2}\mathrm{Re}\left(\alpha_{3\mu\lambda}^{(b)}\right)\right]\\\
\dfrac{1}{2}(1-\delta_{\mu\nu})\left[\mathrm{Re}\left(\alpha_{3\mu\nu}^{(a)}\right)-\mathrm{Re}\left(\alpha_{2\mu\nu}^{(a)}\right)+\mathrm{sym.}\right]\leavevmode\nobreak\
,$ (35)

where ‘sym.’ refers to a swapping between indices $\mu$ and $\nu$, and we have
introduced the following autocorrelation spectra:

$\displaystyle\alpha_{1\mu}\equiv\displaystyle\int_{-\infty}^{0}\mathop{}\\!\mathrm{d}\tau\leavevmode\nobreak\
\left\langle
c_{1\mu}c_{1\mu}^{\tau\star}\right\rangle\exp^{\mathrm{j}\omega_{\mu}\tau}\leavevmode\nobreak\
,$ (36)
$\displaystyle\alpha_{2\mu\nu}^{(a)}\equiv\displaystyle\int_{-\infty}^{0}\mathop{}\\!\mathrm{d}\tau\leavevmode\nobreak\
\left\langle
c_{2\mu\nu}c_{2\nu\mu}^{\tau}\right\rangle\exp^{\mathrm{j}(\omega_{\mu}-\omega_{\nu})\tau}\leavevmode\nobreak\
,$ (37)
$\displaystyle\alpha_{2\mu\nu}^{(b)}\equiv\displaystyle\int_{-\infty}^{0}\mathop{}\\!\mathrm{d}\tau\leavevmode\nobreak\
\left\langle
c_{2\mu\nu}c_{2\mu\nu}^{\tau\star}\right\rangle\exp^{\mathrm{j}(\omega_{\mu}-\omega_{\nu})\tau}\leavevmode\nobreak\
,$ (38)
$\displaystyle\alpha_{2\mu}^{(R)}\equiv\displaystyle\int_{-\infty}^{0}\mathop{}\\!\mathrm{d}\tau\leavevmode\nobreak\
\left\langle\mathrm{Re}\left(c_{2\mu\mu}\right)\mathrm{Re}\left(c_{2\mu\mu}^{\tau}\right)\right\rangle\leavevmode\nobreak\
,$ (39)
$\displaystyle\alpha_{2\mu}^{(I)}\equiv\displaystyle\int_{-\infty}^{0}\mathop{}\\!\mathrm{d}\tau\leavevmode\nobreak\
\left\langle\mathrm{Im}\left(c_{2\mu\mu}\right)\mathrm{Im}\left(c_{2\mu\mu}^{\tau}\right)\right\rangle\leavevmode\nobreak\
,$ (40)
$\displaystyle\alpha_{3\mu\nu}^{(a)}\equiv\displaystyle\int_{-\infty}^{0}\mathop{}\\!\mathrm{d}\tau\leavevmode\nobreak\
\left\langle
c_{3\mu\nu}c_{3\nu\mu}^{\tau\star}\right\rangle\exp^{\mathrm{j}(\omega_{\mu}+\omega_{\nu})\tau}\leavevmode\nobreak\
,$ (41)
$\displaystyle\alpha_{3\mu\nu}^{(b)}\equiv\displaystyle\int_{-\infty}^{0}\mathop{}\\!\mathrm{d}\tau\leavevmode\nobreak\
\left\langle
c_{3\mu\nu}c_{3\mu\nu}^{\tau\star}\right\rangle\exp^{\mathrm{j}(\omega_{\mu}+\omega_{\nu})\tau}\leavevmode\nobreak\
.$ (42)

It can already be seen, without having to specify the stochastic processes
$c_{i}(t)$, that their autocorrelation spectra are independent of the two
stochastic variables $A$ and $\Phi$. Furthermore, under the assumption that
the turbulence characterising the convective motions at the top of the
envelope of solar-like oscillators is stationary, the autocorrelation
functions appearing in the definition of these coefficients are only dependent
on the time increment $\tau$, and not on the absolute time $t$. As such, the
$\alpha_{i}$ are also independent of time $t$. All things considered, they are
therefore simply complex constants.

###  3.3 The equivalent simplified amplitude equations

As we have mentioned above, either the stochastic differential equations (Eqs.
28 and 29) together or the Fokker-Planck equation (Eq. 30), both of which are
equivalent to one another, can be used to model the time evolution of the real
amplitude and phase of the modes. However, both are equally impractical to
use, albeit for different reasons. Numerically integrating the stochastic
equations proves extremely expensive, because a large range of very different
timescales must be resolved. Indeed, the total integration time must far
exceed the typical timescale of the slowly varying mode amplitude. But at the
same time, the rapidly varying phase $\omega t$ appearing in Eqs. 28 and 29
must be accurately resolved. Finally, the whole range of memory timescales
associated with the processes $c_{i}(t)$ must also be resolved, which is
problematic, as it corresponds to the range of timescales in the turbulent
cascade, and is therefore very wide in high Reynolds number flows.
Consequently, the numerical integration of Eqs. 28 and 29 requires an
unreasonably small time step compared to the total integration time.

By contrast, the Fokker-Planck equation (Eq. 30) does not have this timescale
problem. Indeed, in computing its probability fluxes and diffusion
coefficients, we have filtered out all rapidly oscillating features (see
Appendix A for more details), and as such, the numerical integration of the
Fokker-Planck equation does not require these very short timescales to be
resolved. However, it poses other difficulties inherent to the integration of
Fokker-Planck equations in general. Indeed, because the PDF $w$ is a function
of the entire parameter space (i.e. not only of time, but also of the
stochastic variables $A_{\mu}$ and $\Phi_{\mu}$), its numerical integration
would require the discretisation of all three variables. For that reason, it
is usually very impractical to directly integrate the Fokker-Planck equation
in time.

Fortunately, while a given stochastic differential equation is equivalent to a
single Fokker-Planck equation, the opposite is not true. Indeed, there exists
an infinite number of stochastic models for the mode amplitudes $A_{\mu}$ and
phases $\Phi_{\mu}$ that possess the same Fokker-Planck equation (Eq. 30) and
therefore contain the exact same statistical information as Eqs. 28 and 29
while having a much simpler form. They are the simplified amplitude equations
(Stratonovich 1965; see also Buchler et al. 1993)

$\displaystyle\mathop{}\\!\mathrm{d}A_{\mu}=\left(\mathcal{G}_{\mu}-\dfrac{1}{2}{{\sum}}_{\lambda\nu}\dfrac{\partial\mathcal{D}^{1/2}_{\mu\lambda}}{\partial
A_{\nu}}\mathcal{D}^{1/2}_{\lambda\nu}\right)\leavevmode\nobreak\
\mathop{}\\!\mathrm{d}t+{{\sum}}_{\nu}\mathcal{D}^{1/2}_{\mu\nu}\circ\mathop{}\\!\mathrm{d}W_{A\nu}\leavevmode\nobreak\
,$ (43)
$\displaystyle\mathop{}\\!\mathrm{d}\Phi_{\mu}=\left(\mathcal{H}_{\mu}-\dfrac{1}{2}{{\sum}}_{\lambda\nu}\dfrac{\partial\mathcal{F}^{1/2}_{\mu\lambda}}{\partial\Phi_{\nu}}\mathcal{F}^{1/2}_{\lambda\nu}\right)\leavevmode\nobreak\
\mathop{}\\!\mathrm{d}t+{{\sum}}_{\nu}\mathcal{F}^{1/2}_{\mu\nu}\circ\mathop{}\\!\mathrm{d}W_{\Phi\nu}\leavevmode\nobreak\
,$ (44)

where $\mathcal{D}^{1/2}$ and $\mathcal{F}^{1/2}$ are the square-root of the
positive definite matrices222Because the ‘off-diagonal’ matrix $\mathcal{E}$
is zero, the square-root of the blocks $\mathcal{D}$ and $\mathcal{F}$
correspond to the blocks of the square-root of the total diffusion matrix. We
note that this would not be the case if $\mathcal{E}$ had not been zero.
$\mathcal{D}$ and $\mathcal{F}$, and $\mathop{}\\!\mathrm{d}W_{A\nu}$ and
$\mathop{}\\!\mathrm{d}W_{\Phi\nu}$ are the increment over
$\mathop{}\\!\mathrm{d}t$ of independent Wiener processes, that is, Gaussian
processes with zero mean and autocorrelation
$\langle\mathop{}\\!\mathrm{d}W_{A\mu}\mathop{}\\!\mathrm{d}W_{A\nu}\rangle=\langle\mathop{}\\!\mathrm{d}W_{\Phi\mu}\mathop{}\\!\mathrm{d}W_{\Phi\nu}\rangle=\delta_{\mu\nu}\mathop{}\\!\mathrm{d}t$.
The symbol $\circ$ is the differential notation of the Stratonovich stochastic
integral. We note that, in order for the equivalence between the Fokker-Planck
equation (Eq. 30) and the stochastic differential equations (Eqs. 43 and 44)
to be justified, a sufficient condition is that the probability fluxes
$\mathcal{G}_{\nu}$ and $\mathcal{H}_{\nu}$, as well as the elements of the
diffusion matrix $\mathcal{D}_{\mu\nu}$ and $\mathcal{F}_{\mu\nu}$, only
depend on time $t$ and on the variables $A_{\mu}$ and $\Phi_{\mu}$ themselves.
In fact, in the present case, the situation is even simpler, since these
coefficients actually only depend on the real amplitudes $A_{\nu}$.

These equations are much more practical to handle than either the exact
amplitude equations derived in Section 3.1 or the Fokker-Planck equation
derived in Section 3.2, in the sense that they allow us to circumvent the
problems outlined above. Indeed, similarly to the Fokker-Planck equation, all
rapidly oscillating terms have been averaged out, which means that the short
timescales $\sim\omega^{-1}$ need not be resolved. Furthermore, the stochastic
part of these equations now has zero memory – because the effect of the finite
width of the turbulent cascade timescale range has been properly and
rigorously incorporated in the coefficients of the Fokker-Planck equation –
which also drastically reduces the range of timescales that we need to
resolve. In addition, Eqs. 43 and 44 are much easier to integrate numerically
than the corresponding Fokker-Planck equation, as only the time variable needs
to be discretised. The only cost is that a large number of independent
realisations must be integrated in order to reconstruct the moments of the
amplitude and phase of the modes.

##  4 Single-mode case

The final result of Section 3 consists in the coupled simplified amplitude
equations (Eqs. 43 and 44). One important aspect of these equations is the
fact that they are coupled: even in the limit of small amplitudes, where the
modes behave linearly, the non-linear nature of the turbulent convection with
which they are coupled makes it possible to transfer energy from mode to mode,
and for their respective phases to evolve in a dependent manner. This was
already noted by previous studies concerning mode scattering in solar-like
oscillators (e.g Goldreich & Murray 1994) or non-linear mode interactions (e.g
Kumar & Goldreich 1989). This can potentially have an impact on the energy
distribution across the $p$-mode spectrum of the star, and may also impact
their frequency and their lifetime. That being said, as a first application of
the present formalism, we temporarily discard the question of mode coupling
and instead focus on the single-mode case. The coupled-mode case, to which we
briefly return at the end of Section 5, will be the subject of a later paper
in this series.

###  4.1 Simplified amplitude equations in the single-mode case

In the single-mode case, we obtain

$\displaystyle\mathop{}\\!\mathrm{d}A=\left(\mathcal{G}-\dfrac{1}{4}\dfrac{\partial\mathcal{D}}{\partial
A}\right)\leavevmode\nobreak\
\mathop{}\\!\mathrm{d}t+\sqrt{\mathcal{D}}\circ\mathop{}\\!\mathrm{d}W_{A}\leavevmode\nobreak\
,$ (45)
$\displaystyle\mathop{}\\!\mathrm{d}\Phi=\left(\mathcal{H}-\dfrac{1}{4}\dfrac{\partial\mathcal{F}}{\partial\Phi}\right)\leavevmode\nobreak\
\mathop{}\\!\mathrm{d}t+\sqrt{\mathcal{F}}\circ\mathop{}\\!\mathrm{d}W_{\Phi}\leavevmode\nobreak\
,$ (46)

where the probability fluxes and diffusion matrix elements become

$\displaystyle\mathcal{G}=A\left[\kappa+\alpha_{2}^{(R)}+\dfrac{3}{2}\mathrm{Re}\left(\alpha_{3}\right)\right]+\dfrac{\mathrm{Re}\left(\alpha_{1}\right)}{2A}\leavevmode\nobreak\
,$ (47)
$\displaystyle\mathcal{H}=\mathrm{Im}\left(\alpha_{3}\right)\leavevmode\nobreak\
,$ (48)
$\displaystyle\mathcal{D}=A^{2}\left[2\alpha_{2}^{(R)}+\mathrm{Re}\left(\alpha_{3}\right)\right]+\mathrm{Re}\left(\alpha_{1}\right)\leavevmode\nobreak\
,$ (49)
$\displaystyle\mathcal{F}=2\alpha_{2}^{(I)}+\mathrm{Re}\left(\alpha_{3}\right)+\dfrac{\mathrm{Re}\left(\alpha_{1}\right)}{A^{2}}\leavevmode\nobreak\
,$ (50)

and the $\alpha_{i}$ are defined by

$\displaystyle\alpha_{1}\equiv\displaystyle\int_{-\infty}^{0}\mathop{}\\!\mathrm{d}\tau\leavevmode\nobreak\
\left\langle
c_{1}c_{1}^{\tau\star}\right\rangle\exp^{\mathrm{j}\omega\tau}\leavevmode\nobreak\
,$ (51)
$\displaystyle\alpha_{2}^{(R)}\equiv\displaystyle\int_{-\infty}^{0}\mathop{}\\!\mathrm{d}\tau\leavevmode\nobreak\
\left\langle\mathrm{Re}\left(c_{2}\right)\mathrm{Re}\left(c_{2}^{\tau}\right)\right\rangle\leavevmode\nobreak\
,$ (52)
$\displaystyle\alpha_{2}^{(I)}\equiv\displaystyle\int_{-\infty}^{0}\mathop{}\\!\mathrm{d}\tau\leavevmode\nobreak\
\left\langle\mathrm{Im}\left(c_{2}\right)\mathrm{Im}\left(c_{2}^{\tau}\right)\right\rangle\leavevmode\nobreak\
,$ (53)
$\displaystyle\alpha_{3}\equiv\displaystyle\int_{-\infty}^{0}\mathop{}\\!\mathrm{d}\tau\leavevmode\nobreak\
\left\langle
c_{3}c_{3}^{\tau\star}\right\rangle\exp^{2\mathrm{j}\omega\tau}\leavevmode\nobreak\
,$ (54)

$\omega$ now being the angular frequency of the single mode under
consideration. Plugging Eqs. 47, 48, 49 and 50 into Eqs. 45 and 46, we obtain

$\mathop{}\\!\mathrm{d}A=\left(A\left(\kappa+\mathrm{Re}\left(\alpha_{3}\right)\right)+\dfrac{\mathrm{Re}\left(\alpha_{1}\right)}{2A}\right)\leavevmode\nobreak\
\mathop{}\\!\mathrm{d}t\\\
+\left(A^{2}\left[2\alpha_{2}^{(R)}+\mathrm{Re}\left(\alpha_{3}\right)\right]+\mathrm{Re}\left(\alpha_{1}\right)\right)^{1/2}\circ\mathop{}\\!\mathrm{d}W_{A}\leavevmode\nobreak\
,$ (55)
$\mathop{}\\!\mathrm{d}\Phi=\mathrm{Im}\left(\alpha_{3}\right)\leavevmode\nobreak\
\mathop{}\\!\mathrm{d}t+\left(2\alpha_{2}^{(I)}+\mathrm{Re}\left(\alpha_{3}\right)+\dfrac{\mathrm{Re}\left(\alpha_{1}\right)}{A^{2}}\right)^{1/2}\circ\mathop{}\\!\mathrm{d}W_{\Phi}\leavevmode\nobreak\
.$ (56)

Alternatively, we can merge these two equations into a single stochastic
differential equation on the complex amplitude $a=A\exp(\mathrm{j}\Phi)$ of
the mode, which yields333Since the diffusion part of the stochastic
differential equation is interpreted in the Stratonovich sense, the usual
chain rule of differentiation applies.

$\mathop{}\\!\mathrm{d}a=\left[a\left(\kappa+\alpha_{3}\right)+\dfrac{\mathrm{Re}\left(\alpha_{1}\right)}{a^{\star}}\right]\leavevmode\nobreak\
\mathop{}\\!\mathrm{d}t\\\
+a\left(2\alpha_{2}^{(R)}+\mathrm{Re}\left(\alpha_{3}\right)+\dfrac{\mathrm{Re}\left(\alpha_{1}\right)}{|a|^{2}}\right)^{1/2}\circ\mathop{}\\!\mathrm{d}W_{A}\\\
+\mathrm{j}a\left(2\alpha_{2}^{(I)}+\mathrm{Re}\left(\alpha_{3}\right)+\dfrac{\mathrm{Re}\left(\alpha_{1}\right)}{|a|^{2}}\right)^{1/2}\circ\mathop{}\\!\mathrm{d}W_{\Phi}\leavevmode\nobreak\
.$ (57)

While Eqs. 55 and 56 – or, equivalently, Eq. 57 – feature all of the above
autocorrelation spectra, $\alpha_{i}$, the first terms on their respective
right-hand sides (which, as will be discussed in more detail in Sections 4.2
and 4.3, governs the mean evolution of the mode amplitudes) only depend on
$\alpha_{1}$ and $\alpha_{3}$. These two autocorrelation spectra can be
computed by plugging the explicit expressions of $\mathbf{L}_{1}^{s}$ (Eq. 4)
and $\mathbf{L}_{0}$ (Eq. 5) into the expressions for $c_{1}(t)$ and
$c_{3}(t)$ (Eqs. 25 and 26), and then the latter into Eqs. 51 and 54. Since
$c_{1}(t)$ and $c_{3}(t)$ depend on the turbulent velocity field,
$\mathbf{u}_{\text{t}}$, their autocorrelation spectrum can be described in
terms of the autocorrelation spectrum of the turbulent velocity itself. We
introduce

$\mathcal{C}_{\omega,\mathbf{k}}(f_{1}\leavevmode\nobreak\
;\leavevmode\nobreak\
f_{2})\equiv\displaystyle\int_{-\infty}^{0}\mathop{}\\!\mathrm{d}\tau\displaystyle\int\mathop{}\\!\mathrm{d}^{3}\delta\mathbf{x}\leavevmode\nobreak\
\left\langle
f_{1}f_{2}^{\tau,\delta\mathbf{x}}\right\rangle\exp^{\mathrm{j}(\omega\tau-\mathbf{k}\cdot\delta\mathbf{x})}\leavevmode\nobreak\
,$ (58)

where

$f^{\tau,\delta\mathbf{x}}\equiv
f(\mathbf{X}+\delta\mathbf{x},T+\tau)\leavevmode\nobreak\ .$ (59)

Then we can define the second-order turbulent velocity spectrum as

$\phi_{ij}^{(2)}(\mathbf{k},\omega)\equiv\mathcal{C}_{\omega,\mathbf{k}}(u_{\text{t},i}\leavevmode\nobreak\
;\leavevmode\nobreak\ u_{\text{t},j})\leavevmode\nobreak\ ,$ (60)

as well as the following fourth-order spectra

$\displaystyle\phi_{ijkl}^{(4a)}(\mathbf{k},\omega)\equiv\mathcal{C}_{\omega,\mathbf{k}}(u_{\text{t},i}\leavevmode\nobreak\
;\leavevmode\nobreak\
u_{\text{t},j}u_{\text{t},k}u_{\text{t},l})\leavevmode\nobreak\ ,$ (61)
$\displaystyle\phi_{ijkl}^{(4b)}(\mathbf{k},\omega)\equiv\mathcal{C}_{\omega,\mathbf{k}}(u_{\text{t},i}u_{\text{t},j}\leavevmode\nobreak\
;\leavevmode\nobreak\ u_{\text{t},k}u_{\text{t},l})\leavevmode\nobreak\ ,$
(62)
$\displaystyle\phi_{ijklmn}^{(4c)}(\mathbf{k},\omega)\equiv\mathcal{C}_{\omega,\mathbf{k}}(u_{\text{t},i}\partial_{m}u_{\text{t},j}\leavevmode\nobreak\
;\leavevmode\nobreak\
u_{\text{t},k}\partial_{n}u_{\text{t},l})\leavevmode\nobreak\ ,$ (63)
$\displaystyle\phi_{ijklm}^{(4d)}(\mathbf{k},\omega)\equiv\mathcal{C}_{\omega,\mathbf{k}}(u_{\text{t},i}\partial_{m}u_{\text{t},j}\leavevmode\nobreak\
;\leavevmode\nobreak\ u_{\text{t},k}u_{\text{t},l})\leavevmode\nobreak\ .$
(64)

The derivation of the autocorrelation spectra $\alpha_{1}$ and $\alpha_{3}$ is
detailed in Appendix B, and we eventually obtain

$\alpha_{1}=\dfrac{2}{\mathcal{I}}\mathfrak{I}\left(\rho_{0}k_{j}k_{l}\xi_{\mathrm{osc,i}}\xi_{\mathrm{osc,k}}^{\star}\phi_{ijkl}^{(4b)}(\mathbf{k},\omega)\right)\leavevmode\nobreak\
,$ (65)

and

$\displaystyle\alpha_{3}=\dfrac{1}{4\mathcal{I}^{2}}\mathfrak{I}\left(\vphantom{\dfrac{1}{2}}\rho_{0}F_{i}^{(1)}F_{j}^{(1)\star}\phi_{ij}^{(2)}(2\mathbf{k},2\omega)\right.$
$\displaystyle+\rho_{0}F_{ij}^{(2)}F_{kl}^{(2)\star}\phi_{ijkl}^{(4b)}(2\mathbf{k},2\omega)+\rho_{0}F_{ijm}^{(3a)}F_{kln}^{(3a)\star}\phi_{ijklmn}^{(4c)}(2\mathbf{k},2\omega)$
$\displaystyle\left.\vphantom{\dfrac{1}{2}}+2\rho_{0}\leavevmode\nobreak\
\mathrm{Re}\left[F_{i}^{(1)}F_{jkl}^{(3b)\star}\phi_{ijkl}^{(4a)}(2\mathbf{k},2\omega)+F_{ijm}^{(3a)}F_{kl}^{(2)\star}\phi_{ijklm}^{(4d)}(2\mathbf{k},2\omega)\right]\right)\leavevmode\nobreak\
,$ (66)

where we recall that the operator $\mathfrak{I}$ is defined by Eq. 6,
$\mathbf{k}$ is the (space-dependent) wave vector of the mode, and the
functions $F^{(1)}$, $F^{(2)}$, $F^{(3a)}$ and $F^{(3b)}$ are given by444The
subscript j, used as a coordinate index, must not be confused with the in-line
notation $\mathrm{j}$, which refers to the imaginary unit.

$F_{i}^{(1)}=4\mathrm{j}k_{j}\xi_{\mathrm{osc,i}}\xi_{\mathrm{osc,j}}+\mathrm{j}k_{i}\xi_{\mathrm{osc,j}}\xi_{\mathrm{osc,j}}+\dfrac{G_{ij,0}}{\omega}k_{k}\xi_{\mathrm{osc,j}}\xi_{\mathrm{osc,k}}\\\
+\dfrac{\partial
G_{ij}}{\partial\widetilde{u_{k}^{\prime\prime}u_{l}^{\prime\prime}}}\dfrac{\widetilde{u_{k}^{\prime\prime}u_{l}^{\prime\prime}}_{0}}{\omega}k_{m}\xi_{\mathrm{osc,m}}\xi_{\mathrm{osc,j}}+\dfrac{\partial
G_{ij}}{\partial(\partial_{k}\widetilde{u_{l}})}jk_{k}\xi_{\mathrm{osc,j}}\xi_{\mathrm{osc,l}}\\\
+\dfrac{\partial
G_{ij}}{\partial\epsilon}\dfrac{\omega_{t}\widetilde{u_{i}^{\prime\prime}u_{i}^{\prime\prime}}_{0}}{2\omega}k_{m}\xi_{\mathrm{osc,j}}\xi_{\mathrm{osc,m}}\leavevmode\nobreak\
,$ (67) $F_{ij}^{(2)}=\left(\dfrac{\partial
G_{ki}}{\partial\widetilde{u_{j}^{\prime\prime}u_{l}^{\prime\prime}}}+\dfrac{\partial
G_{ki}}{\partial\widetilde{u_{l}^{\prime\prime}u_{j}^{\prime\prime}}}\right)\xi_{\mathrm{osc,l}}\xi_{\mathrm{osc,k}}+\dfrac{\partial
G_{ki}}{\partial\epsilon}\omega_{t}\xi_{\mathrm{osc,j}}\xi_{\mathrm{osc,k}}\leavevmode\nobreak\
,$ (68) $F_{ijk}^{(3a)}=\dfrac{\partial
G_{li}}{\partial(\partial_{k}\widetilde{u_{j}})}\dfrac{1}{\omega}\mathrm{j}k_{m}\xi_{\mathrm{osc,l}}\xi_{\mathrm{osc,m}}$
(69)

and

$F_{ijk}^{(3b)}=-\dfrac{\partial
G_{li}}{\partial\widetilde{u_{j}^{\prime\prime}u_{k}^{\prime\prime}}}\dfrac{1}{\omega}k_{m}\xi_{\mathrm{osc,l}}\xi_{\mathrm{osc,m}}-\dfrac{1}{2}\dfrac{\partial
G_{li}}{\partial\epsilon}\dfrac{\omega_{t}}{\omega}k_{m}\xi_{\mathrm{osc,l}}\xi_{\mathrm{osc,m}}\delta_{jk}\leavevmode\nobreak\
.$ (70)

These four functions can be broken down into several contributions, thus
allowing us to separate the effect of each physical process to the
turbulence–oscillation coupling. In the present case, one can identify the
turbulent pressure contribution, represented by the advection term in the
stochastic velocity equation (see Eq. 26 in Paper I), and which is responsible
for the entirety of $\alpha_{1}$, as well as the first two terms on the right-
hand side of Eq. 67. On the other hand, all the other contributions to
$\alpha_{3}$ stem from the $G_{ij}$ term in the stochastic velocity equation,
which we recall encompasses the collective effect of the buoyancy force, the
fluctuating gas pressure force, and the turbulent dissipation (see Paper I,
and more specifically Appendix A therein, for more details). We also note
that, Eqs. 65 and 66 being written in the form of spatial integrals spanning
across the entire volume of the star, this formalism allows us to determine
which regions of the star are most responsible for the coupling between the
turbulent motions and the oscillatory motions.

###  4.2 Mode driving

Multiplying Eq. 55 by $2A$ and taking the ensemble average, we obtain the
following evolution equation for the mean555It must be understood that in this
context, the word ‘mean’ refers to ensemble average, and not time average. As
such, $E_{m}$ is susceptible to depend on time. energy of the mode
$E_{m}(t)\equiv\langle A(t)^{2}\rangle$:

$\dfrac{\mathop{}\\!\mathrm{d}E_{m}}{\mathop{}\\!\mathrm{d}t}=2E_{m}\left(\vphantom{f_{1}^{\tau}}\kappa+\mathrm{Re}(\alpha_{3})\right)+\mathrm{Re}(\alpha_{1})\leavevmode\nobreak\
.$ (71)

We note that, because $W_{A}(t)$ is a Wiener processes, the diffusion part of
Eq. 55 (i.e. the second, stochastic term on its right-hand side) is
necessarily of zero mean, even when the diffusion coefficient depends
explicitly on the stochastic variables themselves (e.g. Gardiner 1994). The
right-hand side of Eq. 71 contains a linear contribution that corresponds to
the linear damping of the mode and an additive term,
$\mathcal{P}\equiv\mathrm{Re}\left(\alpha_{1}\right)$, which corresponds to
the excitation rate of the mode (i.e the amount of energy injected into the
mode per unit time). We focus on the latter for the moment. It can be seen
that $\mathcal{P}$ only contains contributions from the turbulence-induced
stochastic perturbation of the wave equation (Eq. 12), in the sense that if
$c_{1}(t)$ was identically zero, $\mathcal{P}$ would also be zero. This is in
accordance with the widely acknowledged picture of solar-like oscillations
being stochastically excited by highly turbulent motions of the plasma at the
top of the convection zone.

Using Eq. 54, the excitation rate $\mathcal{P}$ of the mode becomes

$\mathcal{P}=\dfrac{2}{\mathcal{I}}\displaystyle\int\mathop{}\\!\mathrm{d}^{3}\mathbf{X}\leavevmode\nobreak\
\rho_{0}^{2}k_{j}k_{l}\mathrm{Re}\left(\xi_{\mathrm{osc,i}}\xi_{\mathrm{osc,k}}^{\star}\phi_{ijkl}^{(4b)}(\mathbf{k},\omega)\right)\leavevmode\nobreak\
,$ (72)

where $\mathbf{k}$ and $\omega$ denote the wave vector and angular frequency
of the mode. This expression is fully similar to the formulation obtained by
previous models for the excitation of solar-like oscillations (see, for
instance, Samadi & Goupil 2001; Chaplin et al. 2005), which gives further
support to the consistency and validity of the method presented here. First,
it is inversely proportional to the inertia of the mode: the larger the mass
flow entailed by an oscillating mode, the harder it is to get the flow to
actually move. Secondly, it appears that the efficiency of mode driving by
turbulence is directly related to the spectrum of the fourth-order moment of
turbulent velocity. This is in accordance with widely accepted results of past
studies. Finally, the integrand is weighted by a quantity that takes the form
of the product of two different components of the wave vector and two
different components of the modal fluid displacement. Qualitatively, it can be
seen that this is closely related to the square of the mode compressibility
$|\bm{\nabla}\cdot\bm{\xi}_{\mathrm{osc}}|$. In other words, the turbulence
drives the mode much more efficiently in regions where the compressibility of
the mode is high. This is also not surprising, as the turbulent pressure must
actually be able to transfer mechanical work to the mode in order to give it
energy, and mechanical work is only transferable if the flow undergoes
successive compression and dilatation phases.

###  4.3 Mode damping and the modal surface effect

We now turn to the first term on the right-hand side of Eq. 71. Solar-like
oscillations being intrinsically stable, it gives the damping rate of the
mode:

$\eta_{\text{tot}}\equiv-\left(\vphantom{f_{1}^{\tau}}\kappa+\mathrm{Re}(\alpha_{3})\right)\leavevmode\nobreak\
.$ (73)

The total damping rate, $\eta_{\text{tot}}$, is directly related to the
observed linewidth at half maximum $\Gamma_{\text{obs}}$ of the resonant peak
corresponding to the mode in the Fourier power spectrum, through

$\Gamma_{\text{obs}}=\eta_{\text{tot}}/\pi\leavevmode\nobreak\ .$ (74)

In the Sun, $\Gamma_{\text{obs}}$ typically has the order of a few $\mu$Hz,
while the eigenfrequency of the modes is much larger ($\sim
3000\leavevmode\nobreak\ \mu$Hz at the maximum spectral height). Eq. 73 shows
that the total damping rate of the mode – and therefore its observed linewidth
– consists of two different contributions. The first one corresponds to the
non-turbulent, deterministic contribution $\kappa$. It represents the effect
of the non-adiabatic energy exchanges between the $p$-modes and the background
in which they develop. Explicitly modelling $\kappa$ requires the inclusion of
non-adiabatic effects in the model, through an energy equation, which is
outside the scope of this study (see the discussion in Section 5 however). The
second one represents the contribution from turbulence, and encompasses a
whole array of individual physical contributions from turbulent pressure,
turbulent dissipation or convective enthalpy flux for instance. In the
following, we only investigate this second contribution, and which we denote
as $\eta_{\text{turb}}$.

As a side note, we remark that taking
$\mathop{}\\!\mathrm{d}E_{m}/\mathop{}\\!\mathrm{d}t=0$ in Eq. 71 (i.e.
assuming that the transient has died away and that the mode has reached a
stationary state) yields the following expression for the stationary average
energy of the mode:

$E_{\text{stat}}\equiv\dfrac{\mathcal{P}}{2\eta_{\text{tot}}}\leavevmode\nobreak\
.$ (75)

This is in perfect accordance with Eq. 5 of Goldreich & Keeley (1977a) for
instance, and reflects the well-established picture of the equilibrium energy
of the mode resulting from a balance between driving and damping processes.

In parallel, taking the ensemble average of Eq. 56, we obtain the following
simple evolution equation for the mean phase
$\Phi_{m}(t)\equiv\langle\Phi(t)\rangle$:

$\dfrac{\mathop{}\\!\mathrm{d}\Phi_{m}}{\mathop{}\\!\mathrm{d}t}=\mathrm{Im}\left(\alpha_{3}\right)\leavevmode\nobreak\
,$ (76)

which is easily integrated to yield

$\Phi_{m}(t)=\mathrm{Im}(\alpha_{3})\leavevmode\nobreak\
t+\Phi_{m,0}\leavevmode\nobreak\ ,$ (77)

where $\Phi_{m,0}$ is an arbitrary initial average phase. In turn, this yields
the following expression for the average global phase of the mode
$\phi(t)\equiv\omega t+\Phi_{m}(t)$:

$\phi(t)=\left(\vphantom{f_{1}^{\tau}}\omega+\mathrm{Im}(\alpha_{3})\right)t+\Phi_{m,0}\leavevmode\nobreak\
.$ (78)

This amounts to a systematic shift, $\delta\omega$, in the angular frequency
of the mode compared to the angular frequency in the absence of turbulence.
This shift actually represents what is commonly referred to as the modal – or
‘intrinsic’ – part of the surface effect (Balmforth 1992b), that is, the
contribution of turbulent convection not to the equilibrium structure but to
the propagation of the waves themselves.

All in all, the turbulence-induced damping rate and the modal surface effect
turn out to correspond respectively to the real and imaginary parts of the
same complex autocorrelation spectrum $\alpha_{3}$, according to

$\displaystyle\eta_{\text{turb}}=-\mathrm{Re}\left(\alpha_{3}\right)\leavevmode\nobreak\
,$ (79)
$\displaystyle\delta\omega=\mathrm{Im}\left(\alpha_{3}\right)\leavevmode\nobreak\
.$ (80)

This is not quite surprising, as the damping rate and oscillatory frequency of
a mode are themselves simply two sides of the complex eigenvalue associated
with the mode – more specifically, its real and imaginary parts. This is
further illustrated by the fact that, in the first term on the right-hand side
of Eq. 57, the linear contribution (i.e. the one proportional to $a$) involves
directly the complex quantity $\alpha_{3}$. This is a very interesting output
of this model as mode excitation on the one hand and mode damping and the
surface effect on the other hand are usually treated separately. Furthermore,
while mixing-length formalisms allow mode damping rates and the surface effect
to be investigated simultaneously, they must nevertheless be the subject of
separate fitting procedures, owing to the large number of free parameters
involved (see for example Houdek et al. 2017). By contrast, the model
presented here allows us to treat all three aspects – mode excitation, in
addition to mode damping and the surface effect – in the same consistent
framework, and without having to resort to separate procedures.

Using the expression for $\alpha_{3}$ given by Eq. 66, we find

$\displaystyle\eta_{\text{turb}}=-\dfrac{1}{4\mathcal{I}^{2}}\displaystyle\int\mathop{}\\!\mathrm{d}^{3}\mathbf{X}\leavevmode\nobreak\
\rho_{0}^{2}\leavevmode\nobreak\ \mathrm{Re}(\mathcal{F})\leavevmode\nobreak\
,$ (81)
$\displaystyle\delta\omega=\dfrac{1}{4\mathcal{I}^{2}}\displaystyle\int\mathop{}\\!\mathrm{d}^{3}\mathbf{X}\leavevmode\nobreak\
\rho_{0}^{2}\leavevmode\nobreak\ \mathrm{Im}(\mathcal{F})\leavevmode\nobreak\
,$ (82)

where we have defined

$\mathcal{F}\equiv\underbrace{F_{i}^{(1)}F_{j}^{(1)\star}\phi_{ij}^{(2)}(2\mathbf{k},2\omega)}_{\mathrm{second-
order}}\\\
+\underbrace{F_{ij}^{(2)}F_{kl}^{(2)\star}\phi_{ijkl}^{(4b)}(2\mathbf{k},2\omega)}_{\mathrm{fourth-
order}}+\underbrace{F_{ijm}^{(3a)}F_{kln}^{(3a)\star}\phi_{ijkl}^{(4c)}(2\mathbf{k},2\omega)}_{\mathrm{fourth-
order}}\\\ +2\leavevmode\nobreak\
\mathrm{Re}\left(\underbrace{F_{i}^{(1)}F_{jkl}^{(3b)\star}\phi_{ijkl}^{(4a)}(2\mathbf{k},2\omega)}_{\mathrm{fourth-
order}}+\underbrace{F_{ijm}^{(3a)}F_{kl}^{(2)\star}\phi_{ijkl}^{(4d)}(2\mathbf{k},2\omega)}_{\mathrm{fourth-
order}}\right)\leavevmode\nobreak\ ,$ (83)

and we recall that the wave vector, $\mathbf{k}$, and the angular frequency,
$\omega$, are those of the mode under consideration.

Some general results can be drawn from Eqs. 81 and 82. First it can be seen
that both quantities go as $\mathcal{I}^{-2}$: similarly to the excitation
rate $\mathcal{P}$, the higher the mass flow pertaining to the oscillation,
the harder it is to take energy from it. This is due to the fact that the wave
variable $\bm{\xi}_{\mathrm{osc}}$ appears to the fourth power in Eqs. 81 and
82. Since the inertia of the mode depends on the square of
$\bm{\xi}_{\mathrm{osc}}$, the normalisation condition on the eigenfunction
naturally involves the square of the inertia (see Section B.4 for more
details). This is similar to the work integral formulation of the damping rate
(e.g. Samadi et al. 2015), where the wave variables appear to the second
power, and therefore the damping rate goes as $\mathcal{I}^{-1}$ instead. We
note that the reason why the integrals on the right-hand side of Eqs. 81 and
82 involve the fourth power of the wave variables is that we compute the mean
effect of the damping on the mode, which corresponds to the autocorrelation of
the instantaneous damping: since the latter depends on the square of the wave
variables, the autocorrelation depends on their fourth power.

Secondly, while the driving source is a fourth-order quantity in terms of the
turbulent velocity (see Eq. 72 for the excitation rate $\mathcal{P}$), the
damping rate and the modal surface effect, by contrast, have both a second-
order and a fourth-order dependence in terms of the turbulent velocity. These
are explicitly broken down in Eq. 83.

Finally, it is possible to separate the different physical contributions to
the turbulent damping rate $\eta_{\text{turb}}$ and the modal surface effect
$\delta\omega$. To that end, we decomposed the function $F_{i}^{1}$ (defined
by Eq. 67) into three different terms, in the following way:

$F_{i}^{(1)}=F_{i,\text{pt}}^{(1)}+F_{i,\text{G}}^{(1)}+F_{i,\delta\text{G}}^{(1)}\leavevmode\nobreak\
,$ (84)

where

$\displaystyle F_{i,\text{pt}}^{(1)}\equiv
4\mathrm{j}k_{j}\xi_{\mathrm{osc,i}}\xi_{\mathrm{osc,j}}+\mathrm{j}k_{i}\xi_{\mathrm{osc,j}}\xi_{\mathrm{osc,j}}\leavevmode\nobreak\
,$ (85) $\displaystyle
F_{i,\text{G}}^{(1)}\equiv\dfrac{G_{ij,0}}{\omega}k_{k}\xi_{\mathrm{osc,j}}\xi_{\mathrm{osc,k}}\leavevmode\nobreak\
,$ (86) $\displaystyle F_{i,\delta\text{G}}^{(1)}\equiv\dfrac{\partial
G_{ij}}{\partial\widetilde{u_{k}^{\prime\prime}u_{l}^{\prime\prime}}}\dfrac{\widetilde{u_{k}^{\prime\prime}u_{l}^{\prime\prime}}_{0}}{\omega}k_{m}\xi_{\mathrm{osc,m}}\xi_{\mathrm{osc,j}}+\dfrac{\partial
G_{ij}}{\partial(\partial_{k}\widetilde{u_{l}})}\mathrm{j}k_{k}\xi_{\mathrm{osc,j}}\xi_{\mathrm{osc,l}}$
$\displaystyle\hskip 28.45274pt+\dfrac{\partial
G_{ij}}{\partial\epsilon}\dfrac{\omega_{t}\widetilde{u_{i}^{\prime\prime}u_{i}^{\prime\prime}}_{0}}{2\omega}k_{m}\xi_{\mathrm{osc,j}}\xi_{\mathrm{osc,m}}\leavevmode\nobreak\
.$ (87)

The first term represents the effect of turbulent pressure; the second term
represents the joint effect of the pressure-rate-of-strain correlation and the
dissipation of turbulent kinetic energy into heat, both of which are
collectively modelled by means of the drift tensor, $G_{ij}$; and the third
term stems from the variations in the drift tensor, $G_{ij}$, under the
influence of both the turbulence and the oscillations. By contrast, the
functions $F_{ij}^{(2)}$, $F_{ijk}^{(3a)}$, and $F_{ijk}^{(3b)}$, defined by
Eqs. 68, 69 and 70, respectively, entirely originate from the modulation of
the drift tensor. Then Eqs. 81 and 82 can be rewritten in the following way:

$\displaystyle\eta_{\text{turb}}=\eta_{\text{pt}}+\eta_{\text{G}}+\eta_{\delta\text{G}}+\eta_{\text{cross}}\leavevmode\nobreak\
,$ (88)
$\displaystyle\delta\omega=\delta\omega_{\text{pt}}+\delta\omega_{\text{G}}+\delta\omega_{\delta\text{G}}+\delta\omega_{\text{cross}}\leavevmode\nobreak\
,$ (89)

where the various physical contributions to the total damping rate are given
by

$\displaystyle\eta_{\text{pt}}\equiv-\dfrac{1}{4\mathcal{I}^{2}}\displaystyle\int\mathop{}\\!\mathrm{d}^{3}\mathbf{X}\leavevmode\nobreak\
\rho_{0}^{2}\leavevmode\nobreak\
\mathrm{Re}\left(\mathcal{F}_{\text{pt}}\right)\leavevmode\nobreak\ ,$ (90)
$\displaystyle\eta_{\text{G}}\equiv-\dfrac{1}{4\mathcal{I}^{2}}\displaystyle\int\mathop{}\\!\mathrm{d}^{3}\mathbf{X}\leavevmode\nobreak\
\rho_{0}^{2}\leavevmode\nobreak\
\mathrm{Re}\left(\mathcal{F}_{\text{G}}\right)\leavevmode\nobreak\ ,$ (91)
$\displaystyle\eta_{\delta\text{G}}\equiv-\dfrac{1}{4\mathcal{I}^{2}}\displaystyle\int\mathop{}\\!\mathrm{d}^{3}\mathbf{X}\leavevmode\nobreak\
\rho_{0}^{2}\leavevmode\nobreak\
\mathrm{Re}\left(\mathcal{F}_{\delta\text{G}}\right)\leavevmode\nobreak\ ,$
(92)
$\displaystyle\eta_{\text{cross}}\equiv-\dfrac{1}{4\mathcal{I}^{2}}\displaystyle\int\mathop{}\\!\mathrm{d}^{3}\mathbf{X}\leavevmode\nobreak\
\rho_{0}^{2}\leavevmode\nobreak\
\mathrm{Re}\left(\mathcal{F}_{\text{cross}}\right)\leavevmode\nobreak\ ,$ (93)

those of the modal surface effect are given by

$\displaystyle\delta\omega_{\text{pt}}\equiv\dfrac{1}{4\mathcal{I}^{2}}\displaystyle\int\mathop{}\\!\mathrm{d}^{3}\mathbf{X}\leavevmode\nobreak\
\rho_{0}^{2}\leavevmode\nobreak\
\mathrm{Im}\left(\mathcal{F}_{\text{pt}}\right)\leavevmode\nobreak\ ,$ (94)
$\displaystyle\delta\omega_{\text{G}}\equiv\dfrac{1}{4\mathcal{I}^{2}}\displaystyle\int\mathop{}\\!\mathrm{d}^{3}\mathbf{X}\leavevmode\nobreak\
\rho_{0}^{2}\leavevmode\nobreak\
\mathrm{Im}\left(\mathcal{F}_{\text{G}}\right)\leavevmode\nobreak\ ,$ (95)
$\displaystyle\delta\omega_{\delta\text{G}}\equiv\dfrac{1}{4\mathcal{I}^{2}}\displaystyle\int\mathop{}\\!\mathrm{d}^{3}\mathbf{X}\leavevmode\nobreak\
\rho_{0}^{2}\leavevmode\nobreak\
\mathrm{Im}\left(\mathcal{F}_{\delta\text{G}}\right)\leavevmode\nobreak\ ,$
(96)
$\displaystyle\delta\omega_{\text{cross}}\equiv\dfrac{1}{4\mathcal{I}^{2}}\displaystyle\int\mathop{}\\!\mathrm{d}^{3}\mathbf{X}\leavevmode\nobreak\
\rho_{0}^{2}\leavevmode\nobreak\
\mathrm{Im}\left(\mathcal{F}_{\text{cross}}\right)\leavevmode\nobreak\ ,$ (97)

and we have defined

$\displaystyle\mathcal{F}_{\text{pt}}\equiv
F_{i,\text{pt}}^{(1)}F_{j,\text{pt}}^{(1)\star}\leavevmode\nobreak\
\phi_{ij}^{(2)}\leavevmode\nobreak\ ,$ (98)
$\displaystyle\mathcal{F}_{\text{G}}\equiv
F_{i,\text{G}}^{(1)}F_{j,\text{G}}^{(1)\star}\leavevmode\nobreak\
\phi_{ij}^{(2)}\leavevmode\nobreak\ ,$ (99)
$\displaystyle\mathcal{F}_{\delta\text{G}}\equiv
F_{i,\delta\text{G}}^{(1)}F_{j,\delta\text{G}}^{(1)\star}\leavevmode\nobreak\
\phi_{ij}^{(2)}+F_{ij}^{(2)}F_{kl}^{(2)\star}\leavevmode\nobreak\
\phi_{ijkl}^{(4b)}+F_{ijm}^{(3a)}F_{kln}^{(3a)\star}\leavevmode\nobreak\
\phi_{ijkl}^{(4c)}$ $\displaystyle\hskip 14.22636pt+2\leavevmode\nobreak\
\mathrm{Re}\left(F_{i,\delta\text{G}}^{(1)}F_{jkl}^{(3b)\star}\leavevmode\nobreak\
\phi_{ijkl}^{(4a)}+F_{ijm}^{(3a)}F_{kl}^{(2)\star}\leavevmode\nobreak\
\phi_{ijkl}^{(4d)}\right)\leavevmode\nobreak\ ,$ (100)
$\displaystyle\mathcal{F}_{\text{cross}}\equiv\left({\sum}_{\mu,\nu=\text{`pt',
`G',
`}\delta\text{G'}}^{\mu\neq\nu}F_{i,\mu}^{(1)}F_{j,\nu}^{(1)\star}\right)\leavevmode\nobreak\
\phi_{ij}^{(2)}$ $\displaystyle\hskip 14.22636pt+2\leavevmode\nobreak\
\mathrm{Re}\left(F_{i,\text{pt}}^{(1)}F_{jkl}^{(3b)\star}\leavevmode\nobreak\
\phi_{ijkl}^{(4a)}+F_{i,\text{G}}^{(1)}F_{jkl}^{(3b)\star}\leavevmode\nobreak\
\phi_{ijkl}^{(4a)}\right)\leavevmode\nobreak\ .$ (101)

All the functions $F^{(1)}$, $F^{(2)}$ and $F^{(3)}$ are implicitly evaluated
at $\mathbf{X}$, and all the turbulent spectra $\phi^{(2)}$ and $\phi^{(4)}$
are implicitly evaluated at $(2\mathbf{k},2\omega)$, even though we dropped
the notation for the sake of clarity. The contributions labelled ‘pt’, ‘G’,
and ‘$\delta\text{G}$’ have the same meaning as in Eq. 84, and those labelled
‘cross’ are cross terms between these different contributions. This
decomposition shows that the first two contributions only depend on the
second-order turbulent velocity spectrum, while the others depend on both the
second- and fourth-order spectra.

###  4.4 Contributions of turbulent dissipation and redistribution versus
turbulent pressure

In Section 4.3 we derived general expressions for the various physical
contributions to mode damping (see Eqs. 90, 91, 92 and 93) and the modal
surface effect (see Eqs. 94, 95, 96 and 97). The resulting expressions are
valid for any specification of the drift tensor $G_{ij}$, and regardless of
whether the mode is radial or not. While we postpone more realistic
applications of these expressions to a later paper in this series, it is still
worthwhile to examine their output in the following simple case. First, we
assume that the drift tensor is constant, in the sense that it does not depend
on the Reynolds-stress tensor,
$\widetilde{u_{k}^{\prime\prime}u_{l}^{\prime\prime}}$, the mean shear tensor,
$\partial_{k}\widetilde{u_{l}}$, or the turbulent dissipation rate, $\epsilon$

$\dfrac{\partial
G_{ij}}{\partial\widetilde{u_{k}^{\prime\prime}u_{l}^{\prime\prime}}}=\dfrac{\partial
G_{ij}}{\partial(\partial_{k}\widetilde{u_{l}})}=\dfrac{\partial
G_{ij}}{\partial\epsilon}=0\leavevmode\nobreak\ .$ (102)

Physically, this means that the rate at which the different components of the
turbulent motions exchange energy, or are dissipated into heat, is fixed. We
note that this assumption is not very realistic, but that it allows for a
simple illustration of the usefulness of this formalism. However, we do not
specify how the behaviour of these different components compare to each other,
meaning that we do not make any assumption regarding the relative value of the
different components of $G_{ij}$. Secondly, we consider a radial mode, such
that both the displacement eigenfunction and the wave vector of the mode are
radial:

$\displaystyle\bm{\xi}_{\mathrm{osc}}=\xi_{r}\leavevmode\nobreak\
\mathbf{e_{r}}\leavevmode\nobreak\ ,$ (103)
$\displaystyle\mathbf{k}=k_{r}\leavevmode\nobreak\
\mathbf{e_{r}}\leavevmode\nobreak\ .$ (104)

Then Eqs. 90, 91 and 92 reduce to

$\displaystyle\eta_{\text{pt}}\equiv-\dfrac{1}{4\mathcal{I}^{2}}\displaystyle\int\mathop{}\\!\mathrm{d}^{3}\mathbf{X}\leavevmode\nobreak\
25\rho_{0}^{2}k_{r}^{2}\left|\xi_{r}\right|^{4}\mathrm{Re}\left(\phi_{rr}^{(2)}\right)\leavevmode\nobreak\
,$ (105)
$\displaystyle\eta_{\text{G}}\equiv-\dfrac{1}{4\mathcal{I}^{2}}\displaystyle\int\mathop{}\\!\mathrm{d}^{3}\mathbf{X}\leavevmode\nobreak\
\rho_{0}^{2}k_{r}^{2}\left|\xi_{r}\right|^{4}\dfrac{G_{ir}G_{jr}}{\omega^{2}}\mathrm{Re}\left(\phi_{ij}^{(2)}\right)\leavevmode\nobreak\
,$ (106) $\displaystyle\eta_{\delta G}=0\leavevmode\nobreak\ ,$ (107)

and Eqs. 94, 95 and 96 become

$\displaystyle\delta\omega_{\text{pt}}\equiv\dfrac{1}{4\mathcal{I}^{2}}\displaystyle\int\mathop{}\\!\mathrm{d}^{3}\mathbf{X}\leavevmode\nobreak\
25\rho_{0}^{2}k_{r}^{2}\left|\xi_{r}\right|^{4}\mathrm{Im}\left(\phi_{rr}^{(2)}\right)\leavevmode\nobreak\
,$ (108)
$\displaystyle\delta\omega_{\text{G}}\equiv\dfrac{1}{4\mathcal{I}^{2}}\displaystyle\int\mathop{}\\!\mathrm{d}^{3}\mathbf{X}\leavevmode\nobreak\
\rho_{0}^{2}k_{r}^{2}\left|\xi_{r}\right|^{4}\dfrac{G_{ir}G_{jr}}{\omega^{2}}\mathrm{Im}\left(\phi_{ij}^{(2)}\right)\leavevmode\nobreak\
,$ (109) $\displaystyle\delta\omega_{\delta G}=0\leavevmode\nobreak\ .$ (110)

We note that, to keep the discussion simple, we omit the crossed contributions
$\eta_{\text{cross}}$ and $\delta\omega_{\text{cross}}$ in the following. The
only remaining contributions to both the damping rate of the modes and their
modal surface effect are the contribution of turbulent pressure, and the
contribution of the drift tensor $G_{ij}$, which we recall here jointly models
the pressure-rate-of-strain correlations and the turbulent dissipation.

These simplified expressions allow us to roughly estimate which process
contributes predominantly to both the damping rate and the modal surface
effect, depending on the frequency regime. Under the assumption that the
region of the star where the damping of the modes and the surface effects
occur is a thin subsurface layer, then we can write

$\displaystyle\dfrac{\eta_{\text{pt}}}{\eta_{\text{G}}}\equiv
R_{\eta}\sim\dfrac{25\omega^{2}\mathrm{Re}\left(\phi_{rr}^{(2)}\right)}{G_{ir}G_{jr}\mathrm{Re}\left(\phi_{ij}^{(2)}\right)}\leavevmode\nobreak\
,$ (111)
$\displaystyle\dfrac{\delta\omega_{\text{pt}}}{\delta\omega_{\text{G}}}\equiv
R_{\delta\omega}\sim\dfrac{25\omega^{2}\mathrm{Im}\left(\phi_{rr}^{(2)}\right)}{G_{ir}G_{jr}\mathrm{Im}\left(\phi_{ij}^{(2)}\right)}\leavevmode\nobreak\
.$ (112)

Explicitly expanding the index summation in the denominator, we obtain

$G_{ir}G_{jr}\phi_{ij}^{(2)}=2G_{hr}^{2}\phi_{hh}^{(2)}+G_{rr}^{2}\phi_{rr}^{(2)}\leavevmode\nobreak\
,$ (113)

where the index ‘h’ refers to an arbitrary horizontal direction. Then

$\displaystyle
R_{\eta}=\dfrac{25\omega^{2}}{G_{rr}^{2}}\left(\vphantom{\dfrac{1}{2}}1+2\Phi_{G}^{2}\Phi_{u,\eta}\right)^{-1}\leavevmode\nobreak\
,$ (114) $\displaystyle
R_{\delta\omega}=\dfrac{25\omega^{2}}{G_{rr}^{2}}\left(\vphantom{\dfrac{1}{2}}1+2\Phi_{G}^{2}\Phi_{u,\delta\omega}\right)^{-1}\leavevmode\nobreak\
,$ (115)

where the anisotropy factors are defined by

$\displaystyle\Phi_{u,\eta}\equiv\left|\dfrac{\mathrm{Re}\left(\phi_{hh}^{(2)}\right)}{\mathrm{Re}\left(\phi_{rr}^{(2)}\right)}\right|\leavevmode\nobreak\
,$ (116)
$\displaystyle\Phi_{u,\delta\omega}\equiv\left|\dfrac{\mathrm{Im}\left(\phi_{hh}^{(2)}\right)}{\mathrm{Im}\left(\phi_{rr}^{(2)}\right)}\right|\leavevmode\nobreak\
,$ (117)
$\displaystyle\Phi_{G}\equiv\left|\dfrac{G_{rh}}{G_{rr}}\right|\leavevmode\nobreak\
.$ (118)

The anisotropy factors $\Phi_{u,\eta}$ and $\Phi_{u,\delta\omega}$ concern the
turbulent velocity field. By contrast, the anisotropy factor $\Phi_{G}$
concerns the drift tensor $G_{ij}$, and physically represents the ratio
between the rate at which vertical and horizontal turbulent motions exchange
energy, and the rate at which the vertical motions are dissipated into heat.

Regardless of the relative weight of the different components of $G_{ij}$,
their order of magnitude is always $\sim\epsilon/k=\omega_{t}$ (e.g. Pope
1994), where we recall that $\epsilon$ is the turbulent dissipation rate, $k$
is the turbulent kinetic energy, and the ratio between the two defines the
turbulent frequency $\omega_{t}$ already introduced above. We note that in the
present model, $\omega_{t}$ is assumed constant, but not necessarily uniform.
Physically, it represents the lifetime of the energy-bearing eddies in the
turbulent cascade, and corresponds to the autocorrelation timescale of
granulation. Typically, close to the surface of the Sun, this timescale is
$\tau_{\text{conv}}\sim 300-500$ s (e.g. Samadi et al. 2015), in which case
$\omega_{t}\sim 0.01-0.02$ rad.s-1. In particular, in the Simplified Langevin
Model, every diagonal component of the drift tensor is equal to (e.g. Pope
2000)

$G_{ii}=-\left(\dfrac{1}{2}+\dfrac{3}{4}C_{0}\right)\omega_{t}\leavevmode\nobreak\
,$ (119)

where $C_{0}=2.1$ is the Kolmogorov constant. Then, we finally obtain

$R_{\eta/\delta\omega}=\dfrac{25\omega^{2}}{\omega_{t}^{2}}\left(\dfrac{1}{2}+\dfrac{3}{4}C_{0}\right)^{-2}\left(\vphantom{\dfrac{1}{2}}1+2\Phi_{G}^{2}\Phi_{u,\eta/\delta\omega}\right)^{-1}\leavevmode\nobreak\
.$ (120)

In the following, we consider indifferently either one of the two ratios
$R_{\eta}$ or $R_{\delta\omega}$, which we simply denote as $R$. The turbulent
velocity anisotropy factor will indifferently refer to $\Phi_{u,\eta}$ or
$\Phi_{u,\delta\omega}$, which we simply denote as $\Phi_{u}$. It can be
thought of as representing the ratio between the squared horizontal and
vertical turbulent velocities, and can therefore be related to the perhaps
more familiar anisotropy factor $\Phi$ introduced by Gough (1977) through

$\Phi\sim 1+2\Phi_{u}\leavevmode\nobreak\ .$ (121)

In Fig. 1, we plot the ratio $R$ between the two contributions to either mode
damping or the modal surface effect, as a function of mode frequency. While
the turbulent frequency $\omega_{t}$ is fixed (we set it to $\omega_{t}=0.03$
rad.s-1), we vary the two anisotropy factors $\Phi_{u}$ and $\Phi_{G}$ (see
caption). As expected from Eq. 120, the turbulent pressure contribution tends
to dominate for high-frequency modes, while the joint effect of the pressure-
rate-of-strain correlation and of the turbulent dissipation tends to prevail
for low-frequency modes. However, the threshold between these two regimes
drastically depends on $\Phi_{u}$ and $\Phi_{G}$. More specifically, for low
values of $\Phi_{G}$, this threshold is on the low-frequency edge of the solar
damping plateau, around $2$ mHz, and does not depend on $\Phi_{u}$. For higher
values of $\Phi_{G}$ – and in particular for $\Phi_{G}=1$, which corresponds
to a vertical-horizontal turbulent energy redistribution rate similar to the
vertical turbulent dissipation rate –, this threshold is shifted to higher
frequencies, and has a much stronger dependence on $\Phi_{u}$.

Figure 1: Ratio between the two physical contributions to either mode damping
or the modal surface effect (turbulent pressure vs. turbulent dissipation and
redistribution of kinetic energy), as a function of mode frequency. Each panel
corresponds to a fixed value of the drift tensor anisotropy, $\Phi_{G}$, and
the colour code refers to the turbulent velocity anisotropy factor,
$\Phi_{u}$. In particular, $\Phi_{u}=1$ corresponds to the isotropic case, and
$\Phi_{u}=0.5$ corresponds to the anisotropy measured in 3D large-eddy
simulations (e.g. Samadi et al. 2003). The turbulent frequency is set to
$\omega_{t}=0.03$ rad.s-1. The region filled in red corresponds to the solar
damping plateau, and the vertical dashed red line marks the frequency of
maximum spectral height in the Sun, $\nu_{\text{max}}$. The horizontal line
marks the threshold $R=1$: above this line, the turbulent pressure
contribution dominates, and below this line the joint effect of turbulent
dissipation and pressure-rate-of-strain correlation prevails. The vertical
axis is logarithmic.

To better illustrate the effect of varying the model parameters, we plot in
Fig. 2 the same ratio $R$, but this time as a function of the drift tensor
anisotropy factor $\Phi_{G}$ (left panel) and turbulent frequency $\omega_{t}$
(right panel), for a fixed mode frequency $\nu=\nu_{\text{max}}=3$ mHz. These
figures show that, if the turbulent velocity anisotropy factor is fixed to the
value observed in large-eddy simulations (i.e. $\Phi_{u}=0.5$; (e.g. Samadi et
al. 2003)), then the turbulent pressure contribution dominates for
$\Phi_{G}\lesssim 1$ and $\nu_{t}\equiv\omega_{t}/(2\pi)\lesssim 5$ mHz,
respectively. On the other hand, for a higher anisotropy factor $\Phi_{G}$ or
a higher turbulent frequency $\nu_{t}$, the turbulent dissipation plays a much
more important role in both mode damping and the modal surface effect. While
we only considered a simplified case for illustration purposes, this still
tends to show how important it is to account for turbulent dissipation in a
realistic way when modelling the lifetime of the modes, or when trying to
correct the surface effect.

Naturally, this picture is a simplistic one, as it stems primarily from
hypothesis (H7) in Paper I: the reduction of the turbulent cascade to a single
timescale, $\omega_{t}^{-1}$. In a more realistic picture, one can assume that
within the continuous range of turbulent timescales, some eddies act on the
modes through turbulent pressure, and others through turbulent dissipation and
pressure-rate-of-strain correlation, depending on the criteria given by Eq.
120 with $\omega_{t}^{-1}$ being replaced with their lifetime. Nevertheless,
the results showcased in Figs. 1 and 2 illustrate how the formalism presented
in this paper allows us to break down the relative importance of the various
physical contributions to mode damping and the modal surface effect.

(a) $R$ as a function of $\Phi_{G}$

(b) $R$ as a function of $\omega_{t}$

Figure 2: Left: Ratio between the two physical contributions to either mode
damping or the surface effect (turbulent pressure vs. turbulent dissipation
and redistribution of kinetic energy), as a function of the drift tensor
anisotropy factor, $\Phi_{G}$, for a mode of frequency
$\nu=\nu_{\text{max}}=3$ mHz. The turbulent velocity anisotropy is set to
$\Phi_{u}=0.5$ and the turbulent frequency to $\omega_{t}=0.03$ rad.s-1. The
vertical dashed line marks the limit $\Phi_{G}=1$: to the left, the
dissipation of the vertical turbulent motions prevails over the redistribution
of turbulent kinetic energy between horizontal and vertical motions, and vice
versa to the right. The horizontal dashed line is the same as in Fig. 1.
Right: Same ratio, as a function of the turbulent frequency, $\omega_{t}$, for
a mode of frequency $\nu=\nu_{\text{max}}=3$ mHz. The two anisotropy factors
are set to $\Phi_{u}=0.5$ and $\Phi_{G}=1.0$. The horizontal dashed line is
the same as in Fig. 1.

##  5 Conclusions

This series of papers aims to investigate Lagrangian stochastic models of
turbulence for the study of the coupling between solar-like oscillations and
turbulent convection, as an alternative to more traditional methods based on
mixing-length theory for example. In Paper I (Philidet et al. 2021) we
established a linear stochastic wave equation designed to govern the
oscillations while encompassing the full effect of the turbulence thereon. In
this second paper we exploit this wave equation to obtain stochastic equations
that govern the coupled evolution of the complex amplitudes of all the modes
of the system simultaneously. As a first application of these equations, we
consider the single-mode case, which yields explicit and simultaneous
expressions for the excitation rate (Eq. 72), the damping rate (Eq. 81), and
the modal – or ‘intrinsic’ – surface effect (Eq. 82) associated with any
radial or non-radial solar-like $p$-mode, directly as a function of the
statistical properties of the underlying turbulent velocity field.

The expressions for the excitation rate, damping rate, and modal surface
effect provide a useful insight into the physical origin of the energetic
aspects of the solar-like oscillations (at least those stemming from
mechanical work), as well as the discrepancy between observed and theoretical
$p$-mode frequencies. The main conclusions are as follows.

  1. 1.

The seismic observables – amplitudes, linewidths, and the modal surface effect
– all simultaneously arise in a coherent manner from the same formalism.
Furthermore, they are explicitly related to the statistical properties of the
underlying turbulent convection.

  2. 2.

The excitation rate of the mode (Eq. 72) reduces to the same theoretical
formulation obtained by previous studies on the subject (e.g. Samadi & Goupil
2001; Chaplin et al. 2005), which further supports the validity of this
formalism.

  3. 3.

We also recover the fact that the damping rate and modal surface effect (Eqs.
81 and 82) are simply the real and imaginary part of the same complex
quantity. This means that it is necessary to treat them not as separate
phenomena, but as two aspects of the same physical process, which should be
reflected in the way they are theoretically predicted or empirically treated.

  4. 4.

Our formalism allows for the separation of the different physical
contributions to both the damping rate (Eq. 88) and the modal surface effect
(Eq. 89). In the present quasi-adiabatic formalism (i.e. under the assumption
that fluid particles conserve their entropy during their motion), we can
separate the effect of the turbulent pressure from the joint effect of the
pressure-rate-of-strain correlation and that of the turbulent dissipation.

  5. 5.

We apply this formalism to a simple case where the relative importance of
these two contributions (turbulent pressure vs. pressure-rate-of-strain
correlation and turbulent dissipation) can be quantified (Eq. 120) in terms of
1) the turbulent frequency, 2) the degree of anisotropy of the Reynolds-stress
tensor, and 3) the degree of anisotropy of the drift tensor, $G_{ij}$ (i.e.
the ratio between the turbulent energy vertical–horizontal redistribution rate
and its dissipation rate). We show that the turbulent pressure tends to
dominate for high-frequency modes, and that the turbulent dissipation and
pressure-rate-of-strain correlation tend to prevail for low-frequency modes.

We stress that the Lagrangian stochastic model of turbulence from which we
started in Paper I, and which we have continued to adopt in the present study,
was deliberately oversimplified so as to provide a proof-of-concept to
illustrate the validity of this formalism and its relevance for the study of
turbulence–oscillation coupling in the context of solar-like oscillators. In
particular, among the various simplifying assumptions that we adopted in Paper
I, two of them stand out. The first is hypothesis H6, according to which the
individual fluid particles conserve their entropy over time. This hypothesis
is substantially drastic in the context of solar-like oscillations, as the
local thermal timescale actually has the same order of magnitude as the period
of the modes, especially in the superadiabatic region. Therefore, this
approximation needs to be dropped to obtain realistic expressions for the
driving rate, damping rate, and modal surface effect associated with solar-
like oscillations. This can be done from the start by adding a stochastic
differential equation for the internal energy associated with the individual
fluid particles in the Lagrangian stochastic model. The second is hypothesis
H7, according to which the lifetime of the turbulent eddies is assumed to be
constant in time – albeit not necessarily uniform – and is parameterised by
the turbulent frequency, $\omega_{t}$. This is an equally drastic
approximation, as the range of timescales associated with the turbulent eddies
is actually very large due to the very high Reynolds number characteristic of
stellar turbulent convection at the top of the convective envelope. Like the
quasi-adiabatic approximation, obtaining a realistic analytical prescription
for the excitation rate, damping rate, and modal surface effect will require
this assumption to be discarded. This can be done by treating $\omega_{t}$ as
a physical property of each individual fluid particle separately. In doing so,
it would also be necessary to add a stochastic differential equation for the
turbulent frequency associated with a given fluid particle in the Lagrangian
stochastic model from the start. Adding turbulent frequency and internal
energy to the set of wave variables used to describe the solar-like
oscillations is therefore an essential task in the development of this
formalism, and will be the subject of a future paper in this series. While it
does not constitute a conceptual challenge, and while many studies in the
fluid dynamics community have already been devoted to the inclusion of
turbulent frequency in turbulence modelling (e.g. Pope & Chen 1990; Pope
1991), the application to solar-like oscillations is likely to involve many
more theoretical developments and yield much more complex relations: it may
become necessary, at some point, to proceed towards a numerical resolution of
these equations instead.

It will also be necessary, in the long run, to discard the assumption that the
PDF of the turbulent flow is Gaussian. Indeed, the Lagrangian stochastic model
from which we started in Paper I is originally constructed under the implicit
assumption that the equivalent Fokker-Planck equation yields a multivariate
Gaussian PDF. By contrast, as shown by 3D hydrodynamic simulations, stellar
turbulent convection is characterised by a typical structure composed of
upflows and downdrafts, each separately exhibiting Gaussian turbulence, such
that the total flow has a bimodal distribution. Because of the asymmetry
between the upflows and the downdrafts – the latter being colder and more
turbulent than the former – the total distribution is not Gaussian. A possible
solution that would allow the non-Gaussian nature of turbulence to be
accounted for would be to use a two-flow model, where the distribution is
determined by the mean and variance of two Gaussian distributions, or
equivalently by the first four moments of the whole distribution. It is
possible to implement two-flow prescriptions in Lagrangian stochastic models
(see for instance Rodean 1996) by using different sets of stochastic
differential equations to model the evolution of the fluid particles,
depending on whether their vertical velocity is positive or negative and on
whether their temperature is higher or lower than the local mean temperature.
Concretely, this would influence the relation between the turbulent
fluctuations – especially the turbulent velocity – and the perturbations
$\mathcal{L}^{s}$ and $\ket{\Theta}$ to the wave equation. This would
constitute an important step towards making the present approach more
realistic, and its results quantitatively applicable to solar-like oscillators
and thus suitable for comparison with observations.

As a final, important note, we recall that the results presented from Section
4 onwards, and in particular the interpretation of the various terms in the
averaged simplified amplitude equations in terms of mode excitation, mode
damping, and the surface effect, were obtained by neglecting mode coupling,
that is, by reducing the coupled amplitude and phase equations (Eqs. 43 and
44) to two equations on the amplitude and phase of a single mode (Eqs. 55 and
56). While such coupling is deemed negligible in solar-type, main-sequence
stars, where it is usually assumed that modes can be studied in isolation from
one another, this assumption is in fact not warranted by any a priori physical
argument. This is a most important caveat of the findings presented in this
study; in the future, it will be necessary to go beyond the single-mode case,
and instead consider the full, coupled system of equations, especially when it
comes to applying the present formalism to the case of a more evolved star. A
later paper in this series will be dedicated to this task.

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##  Appendix A Derivation of the Fokker-Planck equation

In this appendix, we detail the derivation of the probability fluxes
$\mathcal{G}_{\mu}$ and $\mathcal{H}_{\mu}$ and the components of the
diffusion matrix $\mathcal{D}_{\mu\nu}$, $\mathcal{E}_{\mu\nu}$ and
$\mathcal{F}_{\mu\nu}$ appearing in the joint-amplitude-phase Fokker-Planck
equation (Eq. 30). In order to account for the finite memory time of
$G_{\mu}(A_{\nu},\Phi_{\nu},t)$ and $H_{\mu}(A_{\nu},\Phi_{\nu},t)$ (defined
as the right-hand sides of Eqs. 28 and 29, respectively), Stratonovich (1965)
showed that one can define effective probability fluxes and diffusion
coefficients in the following way:

$\displaystyle\mathcal{G}_{\mu}=\langle
G_{\mu}\rangle+{{\sum}}_{\nu}\left\\{\vphantom{\dfrac{1}{A}}\dfrac{\partial
G_{\mu}}{\partial A_{\nu}}\leavevmode\nobreak\ ;\leavevmode\nobreak\
G_{\nu}^{\tau}\right\\}+{{\sum}}_{\nu}\left\\{\vphantom{\dfrac{1}{A}}\dfrac{\partial
G_{\mu}}{\partial\Phi_{\nu}}\leavevmode\nobreak\ ;\leavevmode\nobreak\
H_{\nu}^{\tau}\right\\}\leavevmode\nobreak\ ,$ (122)
$\displaystyle\mathcal{H}_{\mu}=\langle
H_{\mu}\rangle+{{\sum}}_{\nu}\left\\{\vphantom{\dfrac{1}{A}}\dfrac{\partial
H_{\mu}}{\partial A_{\nu}}\leavevmode\nobreak\ ;\leavevmode\nobreak\
G_{\nu}^{\tau}\right\\}+{{\sum}}_{\nu}\left\\{\vphantom{\dfrac{1}{A}}\dfrac{\partial
H_{\mu}}{\partial\Phi_{\nu}}\leavevmode\nobreak\ ;\leavevmode\nobreak\
H_{\nu}^{\tau}\right\\}\leavevmode\nobreak\ ,$ (123)
$\displaystyle\mathcal{D}_{\mu\nu}=\left\\{\vphantom{\dfrac{1}{A}}G_{\mu}\leavevmode\nobreak\
;\leavevmode\nobreak\
G_{\nu}^{\tau}\right\\}+\left\\{\vphantom{\dfrac{1}{A}}G_{\nu}\leavevmode\nobreak\
;\leavevmode\nobreak\ G_{\mu}^{\tau}\right\\}\leavevmode\nobreak\ ,$ (124)
$\displaystyle\mathcal{E}_{\mu\nu}=\left\\{\vphantom{\dfrac{1}{A}}G_{\mu}\leavevmode\nobreak\
;\leavevmode\nobreak\
H_{\nu}^{\tau}\right\\}+\left\\{\vphantom{\dfrac{1}{A}}H_{\nu}\leavevmode\nobreak\
;\leavevmode\nobreak\ G_{\mu}^{\tau}\right\\}\leavevmode\nobreak\ ,$ (125)
$\displaystyle\mathcal{F}_{\mu\nu}=\left\\{\vphantom{\dfrac{1}{A}}H_{\mu}\leavevmode\nobreak\
;\leavevmode\nobreak\
H_{\nu}^{\tau}\right\\}+\left\\{\vphantom{\dfrac{1}{A}}H_{\nu}\leavevmode\nobreak\
;\leavevmode\nobreak\ H_{\mu}^{\tau}\right\\}\leavevmode\nobreak\ ,$ (126)

where $\langle.\rangle$ denotes an ensemble average, and the bilinear operator
$\\{.\leavevmode\nobreak\ ;\leavevmode\nobreak\ .\\}$ is defined by

$\left\\{f_{1}\leavevmode\nobreak\ ;\leavevmode\nobreak\
f_{2}^{\tau}\right\\}\equiv\displaystyle\int_{-\infty}^{0}\mathop{}\\!\mathrm{d}\tau\leavevmode\nobreak\
\left[\left\langle\vphantom{f_{1}^{\tau}}f_{1}(t)f_{2}(t+\tau)\right\rangle-\left\langle\vphantom{f_{1}^{\tau}}f_{1}(t)\right\rangle\left\langle\vphantom{f_{1}^{\tau}}f_{2}(t+\tau)\right\rangle\right]\leavevmode\nobreak\
.$ (127)

We can rewrite each of the stochastic processes $c_{1\mu}(t)$,
$c_{2\mu\nu}(t)$ and $c_{3\mu\nu}(t)$ in the following polar form:

$\displaystyle c_{1\mu}=C_{1\mu}\exp^{j\phi_{1\mu}}\leavevmode\nobreak\ ,$
(128) $\displaystyle
c_{2\mu\nu}=C_{2\mu\nu}\exp^{j\phi_{2\mu\nu}}\leavevmode\nobreak\ ,$ (129)
$\displaystyle
c_{3\mu\nu}=C_{3\mu\nu}\exp^{j\phi_{3\mu\nu}}\leavevmode\nobreak\ ,$ (130)

where the functions $C_{i}(t)$ and $\phi_{i}(t)$ are both real. Then the
right-hand sides of Eqs. 28 and 29 can be rewritten

$G_{\mu}(A_{\lambda},\Phi_{\lambda},t)=\kappa_{\mu}A_{\mu}+C_{1\mu}\cos\left(\vphantom{f_{1}^{\tau}}\omega_{\mu}t+\Phi_{\mu}-\phi_{1\mu}\right)\\\
+{{\sum}}_{\lambda}A_{\lambda}C_{2\mu\lambda}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\lambda})t+\Phi_{\mu}-\Phi_{\lambda}-\phi_{2\mu\lambda}\right)\\\
+{{\sum}}_{\lambda}A_{\lambda}C_{3\mu\lambda}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\lambda})t+\Phi_{\mu}+\Phi_{\lambda}-\phi_{3\mu\lambda}\right)$
(131)

and

$H_{\mu}(A_{\lambda},\Phi_{\lambda},t)=-\dfrac{1}{A_{\mu}}C_{1\mu}\sin\left(\vphantom{f_{1}^{\tau}}\omega_{\mu}t+\Phi_{\mu}-\phi_{1\mu}\right)\\\
-\dfrac{1}{A_{\mu}}{{\sum}}_{\lambda}A_{\lambda}C_{2\mu\lambda}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\lambda})t+\Phi_{\mu}-\Phi_{\lambda}-\phi_{2\mu\lambda}\right)\\\
-\dfrac{1}{A_{\mu}}{{\sum}}_{\lambda}A_{\lambda}C_{3\mu\lambda}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\lambda})t+\Phi_{\mu}+\Phi_{\lambda}-\phi_{3\mu\lambda}\right)\leavevmode\nobreak\
.$ (132)

Their respective derivatives with respect to $A_{\nu}$ and $\Phi_{\nu}$ are

$\dfrac{\partial G_{\mu}}{\partial
A_{\nu}}=\delta_{\mu\nu}\kappa_{\mu}+C_{2\mu\nu}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\nu})t+\Phi_{\mu}-\Phi_{\nu}-\phi_{2\mu\nu}\right)\\\
+C_{3\mu\nu}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\nu})t+\Phi_{\mu}+\Phi_{\nu}-\phi_{2\mu\nu}\right)\leavevmode\nobreak\
,$ (133) $\dfrac{\partial
G_{\mu}}{\partial\Phi_{\nu}}=-\delta_{\mu\nu}C_{1\mu}\sin\left(\vphantom{f_{1}^{\tau}}\omega_{\mu}t+\Phi_{\mu}-\phi_{1\mu}\right)\\\
-\delta_{\mu\nu}{{\sum}}_{\lambda}A_{\lambda}C_{2\mu\lambda}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\lambda})t+\Phi_{\mu}-\Phi_{\lambda}-\phi_{2\mu\lambda}\right)\\\
+A_{\nu}C_{2\mu\nu}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\nu})t+\Phi_{\mu}-\Phi_{\nu}-\phi_{2\mu\nu}\right)\\\
-\delta_{\mu\nu}{{\sum}}_{\lambda}A_{\lambda}C_{3\mu\lambda}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\lambda})t+\Phi_{\mu}+\Phi_{\lambda}-\phi_{3\mu\lambda}\right)\\\
-A_{\nu}C_{3\mu\nu}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\nu})t+\Phi_{\mu}+\Phi_{\nu}-\phi_{3\mu\nu}\right)\leavevmode\nobreak\
,$ (134) $\dfrac{\partial H_{\mu}}{\partial
A_{\nu}}=\dfrac{\delta_{\mu\nu}}{A_{\mu}^{2}}C_{1\mu}\sin\left(\vphantom{f_{1}^{\tau}}\omega_{\mu}t+\Phi_{\mu}-\phi_{1\mu}\right)\\\
+\dfrac{\delta_{\mu\nu}}{A_{\mu}^{2}}{{\sum}}_{\lambda}A_{\lambda}C_{2\mu\lambda}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\lambda})t+\Phi_{\mu}-\Phi_{\lambda}-\phi_{2\mu\lambda}\right)\\\
-\dfrac{1}{A_{\mu}}C_{2\mu\nu}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\nu})t+\Phi_{\mu}-\Phi_{\nu}-\phi_{2\mu\nu}\right)\\\
+\dfrac{\delta_{\mu\nu}}{A_{\mu}^{2}}{{\sum}}_{\lambda}A_{\lambda}C_{3\mu\lambda}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\lambda})t+\Phi_{\mu}+\Phi_{\lambda}-\phi_{3\mu\lambda}\right)\\\
-\dfrac{1}{A_{\mu}}C_{3\mu\nu}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\nu})t+\Phi_{\mu}+\Phi_{\nu}-\phi_{3\mu\nu}\right)\leavevmode\nobreak\
,$ (135)

and

$\dfrac{\partial
H_{\mu}}{\partial\Phi_{\nu}}=-\dfrac{\delta_{\mu\nu}}{A_{\mu}}C_{1\mu}\cos\left(\vphantom{f_{1}^{\tau}}\omega_{\mu}t+\Phi_{\mu}-\phi_{1\mu}\right)\\\
-\dfrac{\delta_{\mu\nu}}{A_{\mu}}{{\sum}}_{\lambda}A_{\lambda}C_{2\mu\lambda}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\lambda})t+\Phi_{\mu}-\Phi_{\lambda}-\phi_{2\mu\lambda}\right)\\\
+\dfrac{A_{\nu}}{A_{\mu}}C_{2\mu\nu}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\nu})t+\Phi_{\mu}-\Phi_{\nu}-\phi_{2\mu\nu}\right)\\\
-\dfrac{\delta_{\mu\nu}}{A_{\mu}}{{\sum}}_{\lambda}A_{\lambda}C_{3\mu\lambda}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\lambda})t+\Phi_{\mu}+\Phi_{\lambda}-\phi_{3\mu\lambda}\right)\\\
-\dfrac{A_{\nu}}{A_{\mu}}C_{3\mu\nu}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\nu})t+\Phi_{\mu}+\Phi_{\nu}-\phi_{3\mu\nu}\right)\leavevmode\nobreak\
.$ (136)

Plugging these into Eqs. 122, 123, 124, 125 and 126, it may seem at first
glance that the expansion of the bilinear operators $\\{\leavevmode\nobreak\
...\leavevmode\nobreak\ ;\leavevmode\nobreak\ ...\leavevmode\nobreak\ \\}$
involves a large number of terms, all of which can be put in the following
general form

$\left\\{\vphantom{\dfrac{1}{A}}C_{\alpha}\cos\left(\vphantom{f_{1}^{\tau}}\Omega_{\alpha}t+\phi_{\alpha}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\
C_{\beta}^{\tau}\cos\left(\vphantom{f_{1}^{\tau}}\Omega_{\beta}(t+\tau)+\phi_{\beta}^{\tau}\right)\right\\}\leavevmode\nobreak\
,$ (137)

with various amplitudes $C_{\alpha/\beta}$, angular frequencies
$\Omega_{\alpha/\beta}$ and phases $\phi_{\alpha/\beta}$ (we note that sines
can always be written as cosines by redefining their phase). Plugging this
general form into Eq. 127, we find

$\displaystyle\left\\{\vphantom{\dfrac{1}{A}}C_{\alpha}\cos\left(\vphantom{f_{1}^{\tau}}\Omega_{\alpha}t+\phi_{\alpha}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\
C_{\beta}^{\tau}\cos\left(\vphantom{f_{1}^{\tau}}\Omega_{\beta}(t+\tau)+\phi_{\beta}^{\tau}\right)\right\\}$
$\displaystyle=\displaystyle\int_{-\infty}^{0}\mathop{}\\!\mathrm{d}\tau\leavevmode\nobreak\
\left\langle
C_{\alpha}C_{\beta}^{\tau}\cos\left(\vphantom{f_{1}^{\tau}}\Omega_{\alpha}t+\phi_{\alpha}\right)\cos\left(\vphantom{f_{1}^{\tau}}\Omega_{\beta}(t+\tau)+\phi_{\beta}^{\tau}\right)\right\rangle$
$\displaystyle=\dfrac{1}{2}\displaystyle\int_{-\infty}^{0}\mathop{}\\!\mathrm{d}\tau\leavevmode\nobreak\
\left\langle
C_{\alpha}C_{\beta}^{\tau}\cos\left(\vphantom{f_{1}^{\tau}}(\Omega_{\alpha}+\Omega_{\beta})t+\Omega_{\beta}\tau+\phi_{\alpha}+\phi_{\beta}^{\tau}\right)\right\rangle$
$\displaystyle\hskip
14.22636pt+\dfrac{1}{2}\displaystyle\int_{-\infty}^{0}\mathop{}\\!\mathrm{d}\tau\leavevmode\nobreak\
\left\langle
C_{\alpha}C_{\beta}^{\tau}\cos\left(\vphantom{f_{1}^{\tau}}(-\Omega_{\alpha}+\Omega_{\beta})t+\Omega_{\beta}\tau-\phi_{\alpha}+\phi_{\beta}^{\tau}\right)\right\rangle\leavevmode\nobreak\
.$ (138)

The first term oscillates in time $t$ with an angular frequency
$\Omega_{\beta}+\Omega_{\alpha}$, while the second one oscillates with an
angular frequency $\Omega_{\beta}-\Omega_{\alpha}$. However, since we are
interested in the long-term variations in the complex amplitude of the modes,
which occur on timescales much larger than $\omega_{\lambda}^{-1}$, all terms
oscillating at frequencies comparable to any $\omega_{\lambda}$ can be
averaged out. This means that Eq. 138 only yields non-vanishing contributions
if either $\Omega_{\alpha}=\Omega_{\beta}$ or
$\Omega_{\alpha}=-\Omega_{\beta}$. By redefining the phase inside the cosine,
we can always restrict ourselves to the first case. Otherwise stated, this
means that the only terms that can ‘interact’ through the operator
$\\{\leavevmode\nobreak\ ...\leavevmode\nobreak\ ;\leavevmode\nobreak\
...\leavevmode\nobreak\ \\}$ to yield a non-oscillating contribution are those
that share the same frequency in the first place. This can happen either
because the two frequencies are rigorously identical or because there is an
accidental resonance of the type $\sum_{i}n_{i}\omega_{i}=0$, where the sum
concerns two or more modes, and $n_{i}$ are integers. In the following we
refer to the first kind of contribution as ‘cross terms’ and to the second
kind as ‘resonances’. We treat each kind separately in the following two
sections.

###  A.1 Cross terms

The contributions from the cross terms in the curly bracket operators
appearing on the right-hand sides of Eqs. 122, 123, 124, 125 and 126 are given
by Eqs. 140, 141, 142, 143, 144 and 145 below666In addition, we also have
$\mathcal{E}_{\mu\nu}=0$. Indeed, all the integrands appearing in
$\\{G_{\mu};H_{\nu}^{\tau}\\}$ are odd functions of $\tau$. Therefore, since
$\\{H_{\nu};G_{\mu}^{\tau}\\}=\\{G_{\mu};H_{\nu}^{-\tau}\\}$, we simply obtain
$\mathcal{E}_{\mu\nu}=\\{G_{\mu};H_{\nu}^{\tau}\\}+\\{G_{\mu};H_{\nu}^{-\tau}\\}=0\leavevmode\nobreak\
.$ (139) . It can be seen that in each of the individual curly brackets
therein, the frequencies in both cosines/sines are indeed rigorously
identical.

$\left\\{\vphantom{\dfrac{1}{A}}\dfrac{\partial G_{\mu}}{\partial
A_{\nu}}\leavevmode\nobreak\ ;\leavevmode\nobreak\
G_{\nu}^{\tau}\right\\}=\left\\{\vphantom{\dfrac{1}{A}}C_{2\mu\nu}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\nu})t+\Phi_{\mu}-\Phi_{\nu}-\phi_{2\mu\nu}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\
A_{\mu}C_{2\nu\mu}^{\tau}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\nu})(t+\tau)+\Phi_{\mu}-\Phi_{\nu}+\phi_{2\nu\mu}^{\tau}\right)\right\\}\\\
+\left\\{\vphantom{\dfrac{1}{A}}C_{3\mu\nu}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\nu})t+\Phi_{\mu}+\Phi_{\nu}-\phi_{3\mu\nu}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\
A_{\mu}C_{3\nu\mu}^{\tau}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\nu})(t+\tau)+\Phi_{\mu}+\Phi_{\nu}-\phi_{3\nu\mu}^{\tau}\right)\right\\}\leavevmode\nobreak\
,$ (140)

$\left\\{\vphantom{\dfrac{1}{A}}\dfrac{\partial
G_{\mu}}{\partial\Phi_{\nu}}\leavevmode\nobreak\ ;\leavevmode\nobreak\
H_{\nu}^{\tau}\right\\}=\left\\{\vphantom{\dfrac{1}{A}}-\delta_{\mu\nu}C_{1\mu}\sin\left(\vphantom{f_{1}^{\tau}}\omega_{\mu}t+\Phi_{\mu}-\phi_{1\mu}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\
-\dfrac{\delta_{\mu\nu}}{A_{\mu}}C_{1\mu}^{\tau}\sin\left(\vphantom{f_{1}^{\tau}}\omega_{\mu}(t+\tau)+\Phi_{\mu}-\phi_{1\mu}^{\tau}\right)\right\\}\\\
+{{\sum}}_{\lambda}\left\\{\vphantom{\dfrac{1}{A}}-\delta_{\mu\nu}A_{\lambda}C_{2\mu\lambda}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\lambda})t+\Phi_{\mu}-\Phi_{\lambda}-\phi_{2\mu\lambda}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\
-\dfrac{A_{\lambda}}{A_{\mu}}C_{2\mu\lambda}^{\tau}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\lambda})(t+\tau)+\Phi_{\mu}-\Phi_{\lambda}-\phi_{2\mu\lambda}^{\tau}\right)\right\\}\\\
+\left\\{\vphantom{\dfrac{1}{A}}A_{\nu}C_{2\mu\nu}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\nu})t+\Phi_{\mu}-\Phi_{\nu}-\phi_{2\mu\nu}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\
\dfrac{A_{\mu}}{A_{\nu}}C_{2\nu\mu}^{\tau}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\nu})(t+\tau)+\Phi_{\mu}-\Phi_{\nu}+\phi_{2\nu\mu}^{\tau}\right)\right\\}\\\
+{{\sum}}_{\lambda}\left\\{\vphantom{\dfrac{1}{A}}-\delta_{\mu\nu}A_{\lambda}C_{3\mu\lambda}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\lambda})t+\Phi_{\mu}+\Phi_{\lambda}-\phi_{3\mu\lambda}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\
-\dfrac{A_{\lambda}}{A_{\mu}}C_{3\mu\lambda}^{\tau}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\lambda})(t+\tau)+\Phi_{\mu}+\Phi_{\lambda}-\phi_{3\mu\lambda}^{\tau}\right)\right\\}\\\
+\left\\{\vphantom{\dfrac{1}{A}}-A_{\nu}C_{3\mu\nu}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\nu})t+\Phi_{\mu}+\Phi_{\nu}-\phi_{3\mu\nu}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\
-\dfrac{A_{\mu}}{A_{\nu}}C_{3\nu\mu}^{\tau}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\nu})(t+\tau)+\Phi_{\mu}+\Phi_{\nu}-\phi_{3\nu\mu}^{\tau}\right)\right\\}\leavevmode\nobreak\
,$ (141)

$\left\\{\vphantom{\dfrac{1}{A}}\dfrac{\partial H_{\mu}}{\partial
A_{\nu}}\leavevmode\nobreak\ ;\leavevmode\nobreak\
G_{\nu}^{\tau}\right\\}=\left\\{\vphantom{\dfrac{1}{A}}\dfrac{\delta_{\mu\nu}}{A_{\mu}^{2}}C_{1\mu}\sin\left(\vphantom{f_{1}^{\tau}}\omega_{\mu}t+\Phi_{\mu}-\phi_{1\mu}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\
\delta_{\mu\nu}C_{1\mu}^{\tau}\cos\left(\vphantom{f_{1}^{\tau}}\omega_{\mu}(t+\tau)+\Phi_{\mu}-\phi_{1\mu}^{\tau}\right)\right\\}\\\
+{{\sum}}_{\lambda}\left\\{\vphantom{\dfrac{1}{A}}\dfrac{\delta_{\mu\nu}}{A_{\mu}^{2}}A_{\lambda}C_{2\mu\lambda}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\lambda})t+\Phi_{\mu}-\Phi_{\lambda}-\phi_{2\mu\lambda}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\
A_{\lambda}C_{2\mu\lambda}^{\tau}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\lambda})(t+\tau)+\Phi_{\mu}-\Phi_{\lambda}-\phi_{2\mu\lambda}^{\tau}\right)\right\\}\\\
+\left\\{\vphantom{\dfrac{1}{A}}-\dfrac{1}{A_{\mu}}C_{2\mu\nu}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\nu})t+\Phi_{\mu}-\Phi_{\nu}-\phi_{2\mu\nu}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\
A_{\mu}C_{2\nu\mu}^{\tau}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\nu})(t+\tau)+\Phi_{\mu}-\Phi_{\nu}+\phi_{2\nu\mu}^{\tau}\right)\right\\}\\\
+{{\sum}}_{\lambda}\left\\{\vphantom{\dfrac{1}{A}}\dfrac{\delta_{\mu\nu}}{A_{\mu}^{2}}A_{\lambda}C_{3\mu\lambda}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\lambda})t+\Phi_{\mu}+\Phi_{\lambda}-\phi_{3\mu\lambda}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\
A_{\lambda}C_{3\mu\lambda}^{\tau}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\lambda})(t+\tau)+\Phi_{\mu}+\Phi_{\lambda}-\phi_{3\mu\lambda}^{\tau}\right)\right\\}\\\
+\left\\{\vphantom{\dfrac{1}{A}}-\dfrac{1}{A_{\mu}}C_{3\mu\nu}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\nu})t+\Phi_{\mu}+\Phi_{\nu}-\phi_{3\mu\nu}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\
A_{\mu}C_{3\nu\mu}^{\tau}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\nu})(t+\tau)+\Phi_{\mu}+\Phi_{\nu}-\phi_{3\nu\mu}^{\tau}\right)\right\\}\leavevmode\nobreak\
,$ (142)

$\left\\{\vphantom{\dfrac{1}{A}}\dfrac{\partial
H_{\mu}}{\partial\Phi_{\nu}}\leavevmode\nobreak\ ;\leavevmode\nobreak\
H_{\nu}^{\tau}\right\\}=\left\\{\vphantom{\dfrac{1}{A}}-\dfrac{\delta_{\mu\nu}}{A_{\mu}}C_{1\mu}\cos\left(\vphantom{f_{1}^{\tau}}\omega_{\mu}t+\Phi_{\mu}-\phi_{1\mu}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\
-\dfrac{\delta_{\mu\nu}}{A_{\mu}}C_{1\mu}^{\tau}\sin\left(\vphantom{f_{1}^{\tau}}\omega_{\mu}(t+\tau)+\Phi_{\mu}-\phi_{1\mu}^{\tau}\right)\right\\}\\\
+{{\sum}}_{\lambda}\left\\{\vphantom{\dfrac{1}{A}}-\dfrac{\delta_{\mu\nu}}{A_{\mu}}A_{\lambda}C_{2\mu\lambda}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\lambda})t+\Phi_{\mu}-\Phi_{\lambda}-\phi_{2\mu\lambda}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\
-\dfrac{A_{\lambda}}{A_{\mu}}C_{2\mu\lambda}^{\tau}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\lambda})(t+\tau)+\Phi_{\mu}-\Phi_{\lambda}-\phi_{2\mu\lambda}^{\tau}\right)\right\\}\\\
+\left\\{\vphantom{\dfrac{1}{A}}\dfrac{A_{\nu}}{A_{\mu}}C_{2\mu\nu}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\nu})t+\Phi_{\mu}-\Phi_{\nu}-\phi_{2\mu\nu}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\
\dfrac{A_{\mu}}{A_{\nu}}C_{2\nu\mu}^{\tau}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\nu})(t+\tau)+\Phi_{\mu}-\Phi_{\nu}+\phi_{2\nu\mu}^{\tau}\right)\right\\}\\\
+{{\sum}}_{\lambda}\left\\{\vphantom{\dfrac{1}{A}}-\dfrac{\delta_{\mu\nu}}{A_{\mu}}A_{\lambda}C_{3\mu\lambda}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\lambda})t+\Phi_{\mu}+\Phi_{\lambda}-\phi_{3\mu\lambda}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\
-\dfrac{A_{\lambda}}{A_{\mu}}C_{3\mu\lambda}^{\tau}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\lambda})(t+\tau)+\Phi_{\mu}+\Phi_{\lambda}-\phi_{3\mu\lambda}^{\tau}\right)\right\\}\\\
+\left\\{\vphantom{\dfrac{1}{A}}-\dfrac{A_{\nu}}{A_{\mu}}C_{3\mu\nu}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\nu})t+\Phi_{\mu}+\Phi_{\nu}-\phi_{3\mu\nu}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\
-\dfrac{A_{\mu}}{A_{\nu}}C_{3\nu\mu}^{\tau}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\nu})(t+\tau)+\Phi_{\mu}+\Phi_{\nu}-\phi_{3\nu\mu}^{\tau}\right)\right\\}\leavevmode\nobreak\
,$ (143)

$\left\\{\vphantom{\dfrac{1}{A}}G_{\mu}\leavevmode\nobreak\
;\leavevmode\nobreak\
G_{\nu}^{\tau}\right\\}=\left\\{\vphantom{\dfrac{1}{A}}C_{1\mu}\cos\left(\vphantom{f_{1}^{\tau}}\omega_{\mu}t+\Phi_{\mu}-\phi_{1\mu}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\
\delta_{\mu\nu}C_{1\mu}^{\tau}\cos\left(\vphantom{f_{1}^{\tau}}\omega_{\mu}(t+\tau)+\Phi_{\mu}-\phi_{1\mu}^{\tau}\right)\right\\}\\\
+\delta_{\mu\nu}{{\sum}}_{\lambda}\left\\{\vphantom{\dfrac{1}{A}}A_{\lambda}C_{2\mu\lambda}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\lambda})t+\Phi_{\mu}-\Phi_{\lambda}-\phi_{2\mu\lambda}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\
A_{\lambda}C_{2\mu\lambda}^{\tau}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\lambda})(t+\tau)+\Phi_{\mu}-\Phi_{\lambda}-\phi_{2\mu\lambda}^{\tau}\right)\right\\}\\\
+(1-\delta_{\mu\nu})\left\\{\vphantom{\dfrac{1}{A}}A_{\nu}C_{2\mu\nu}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\nu})t+\Phi_{\mu}-\Phi_{\nu}-\phi_{2\mu\nu}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\
A_{\mu}C_{2\nu\mu}^{\tau}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\nu})(t+\tau)+\Phi_{\mu}-\Phi_{\nu}+\phi_{2\nu\mu}^{\tau}\right)\right\\}\\\
+\delta_{\mu\nu}{{\sum}}_{\lambda}\left\\{\vphantom{\dfrac{1}{A}}A_{\lambda}C_{3\mu\lambda}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\lambda})t+\Phi_{\mu}+\Phi_{\lambda}-\phi_{3\mu\lambda}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\
A_{\lambda}C_{3\mu\lambda}^{\tau}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\lambda})(t+\tau)+\Phi_{\mu}+\Phi_{\lambda}-\phi_{3\mu\lambda}^{\tau}\right)\right\\}\\\
+(1-\delta_{\mu\nu})\left\\{\vphantom{\dfrac{1}{A}}A_{\nu}C_{3\mu\nu}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\nu})t+\Phi_{\mu}+\Phi_{\nu}-\phi_{3\mu\nu}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\
A_{\mu}C_{3\nu\mu}^{\tau}\cos\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}+\omega_{\nu})(t+\tau)+\Phi_{\mu}+\Phi_{\nu}-\phi_{3\nu\mu}^{\tau}\right)\right\\}\leavevmode\nobreak\
,$ (144)

$\left\\{\vphantom{\dfrac{1}{A}}H_{\mu}\leavevmode\nobreak\
;\leavevmode\nobreak\
H_{\nu}^{\tau}\right\\}=\left\\{\vphantom{\dfrac{1}{A}}-\dfrac{1}{A_{\mu}}C_{1\mu}\sin\left(\vphantom{f_{1}^{\tau}}\omega_{\mu}t+\Phi_{\mu}-\phi_{1\mu}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\
-\dfrac{\delta_{\mu\nu}}{A_{\mu}}C_{1\mu}^{\tau}\sin\left(\vphantom{f_{1}^{\tau}}\omega_{\mu}(t+\tau)+\Phi_{\mu}-\phi_{1\mu}^{\tau}\right)\right\\}\\\
+\delta_{\mu\nu}{{\sum}}_{\lambda}\left\\{\vphantom{\dfrac{1}{A}}-\dfrac{A_{\lambda}}{A_{\mu}}C_{2\mu\lambda}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\lambda})t+\Phi_{\mu}-\Phi_{\lambda}-\phi_{2\mu\lambda}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\
-\dfrac{A_{\lambda}}{A_{\mu}}C_{2\mu\lambda}^{\tau}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\lambda})(t+\tau)+\Phi_{\mu}-\Phi_{\lambda}-\phi_{2\mu\lambda}^{\tau}\right)\right\\}\\\
+(1-\delta_{\mu\nu})\left\\{\vphantom{\dfrac{1}{A}}-\dfrac{A_{\nu}}{A_{\mu}}C_{2\mu\nu}\sin\left(\vphantom{f_{1}^{\tau}}(\omega_{\mu}-\omega_{\nu})t+\Phi_{\mu}-\Phi_{\nu}-\phi_{2\mu\nu}\right)\leavevmode\nobreak\
;\leavevmode\nobreak\