Theory of solar oscillations in the inertial frequency range: Linear modes of the convection zone
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A&A 662, A16 (2022)
https://doi.org/10.1051/0004-6361/202243164 Astronomy
c Y. Bekki et al. 2022 &
Astrophysics
Theory of solar oscillations in the inertial frequency range: Linear
modes of the convection zone
Yuto Bekki1 , Robert H. Cameron1, and Laurent Gizon1,2,3
1
Max-Planck-Institut für Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany
e-mail: bekki@mps.mpg.de
2
Institut für Astrophysik, Georg-August-Universtät Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
3
Center for Space Science, NYUAD Institute, New York University Abu Dhabi, Abu Dhabi, UAE
Received 20 January 2022 / Accepted 9 March 2022
ABSTRACT
Context. Several types of global-scale inertial modes of oscillation have been observed on the Sun. These include the equatorial
Rossby modes, critical-latitude modes, and high-latitude modes. However, the columnar convective modes (predicted by simulations
and also known as banana cells or thermal Rossby waves) remain elusive.
Aims. We aim to investigate the influence of turbulent diffusivities, non-adiabatic stratification, differential rotation, and a latitudinal
entropy gradient on the linear global modes of the rotating solar convection zone.
Methods. We numerically solved for the eigenmodes of a rotating compressible fluid inside a spherical shell. The model takes into ac-
count the solar stratification, turbulent diffusivities, differential rotation (determined by helioseismology), and the latitudinal entropy
gradient. As a starting point, we restricted ourselves to a superadiabaticity and turbulent diffusivities that are uniform in space. We
identified modes in the inertial frequency range, including the columnar convective modes as well as modes of a mixed character. The
corresponding mode dispersion relations and eigenfunctions are computed for azimuthal orders of m ≤ 16.
Results. The three main results are as follows. Firstly, we find that, for m & 5, the radial dependence of the equatorial Rossby modes
with no radial node (n = 0) is radically changed from the traditional expectation (rm ) for turbulent diffusivities &1012 cm2 s−1 . Sec-
ondly, we find mixed modes, namely, modes that share properties of the equatorial Rossby modes with one radial node (n = 1) and the
columnar convective modes, which are not substantially affected by turbulent diffusion. Thirdly, we show that the m = 1 high-latitude
mode in the model is consistent with the solar observations when the latitudinal entropy gradient corresponding to a thermal wind
balance is included (baroclinically unstable mode).
Conclusions. To our knowledge, this work is the first realistic eigenvalue calculation of the global modes of the rotating solar con-
vection zone. This calculation reveals a rich spectrum of modes in the inertial frequency range, which can be directly compared to the
observations. In turn, the observed modes can inform us about the solar convection zone.
Key words. convection – Sun: interior – Sun: rotation
1. Introduction The first family of inertial modes observed on the
Sun consists of the quasi-toroidal equatorial Rossby modes
Based on ten years of observations from the Helioseismic and (Löptien et al. 2018). They are analogous to the sectoral r
Magnetic Imager (HMI) onboard the Solar Dynamics Obser- modes described by, for instance, Papaloizou & Pringle (1978),
vatory (SDO), Gizon et al. (2021) discovered that the Sun sup- Smeyers et al. (1981), and Saio (1982). On the Sun, these modes
ports a large number of global modes of inertial oscillations. The have 3 ≤ m ≤ 15 with a well-defined dispersion relation close to
restoring force for these inertial modes is the Coriolis force and ω = −2Ωref /(m + 1), where ω is the mode angular frequency and
thus the modes have periods comparable to the solar rotation Ωref /2π = 453.1 nHz is the equatorial rotation rate at the surface.
period (∼27 days). The inertial modes can potentially be used as For positive values of m, a negative ω indicates retrograde prop-
a tool to probe the interior of the Sun because they are sensi- agation. There have been several follow-up studies that confirm
tive to properties of the deep convection zone that the p modes these observations (e.g., Liang et al. 2019; Hanasoge & Mandal
are insensitive to. In order to achieve this goal, we need a better 2019; Proxauf et al. 2020; Hanson et al. 2020). Using a one-
understanding of the mode physics. dimensional β-plane model with a parabolic shear flow and vis-
The low frequency modes of solar oscillation have been cosity, Gizon et al. (2020a) showed that these modes, among
described in a rotating frame (angular velocity Ωref ). Because others, are affected by differential rotation and are trapped
the Sun is essentially symmetric about its rotation axis, the between the critical latitudes where the phase speed of a mode
velocity of each mode in the rotating frame has the form of is equal to the local rotational velocity. Fournier et al. (2022)
u(r, θ) exp [i(mφ − ωt)], where r is the radius, θ is the colatitude, extended this model to a spherical geometry using a realistic dif-
φ is the longitude, m is the azimuthal order, and ω is the mode ferential rotation model and found that some Rossby modes can
eigenfrequency. Gizon et al. (2021) provide all observed eigen- be unstable for m ≤ 3.
frequencies ω for each m, along with the eigenfunctions (vθ and Gizon et al. (2021) also report a family of modes at mid-
vφ at the surface) for a few selected modes. latitudes that are localized near their critical latitudes. Several
A16, page 1 of 23
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0),
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Open Access funding provided by Max Planck Society.
A&A 662, A16 (2022)
tens of critical-latitude modes have been identified in the Firstly, here we show that when the turbulent viscosity is
range m ≤ 10. Another family of inertial modes introduced above approximately 1012 cm2 s−1 , the equatorial Rossby modes
by the Sun’s differential rotation are the high-latitude modes with no radial node (n = 0) strongly depart from the expected
(Gizon et al. 2021). The highest amplitude mode (∼10−20 m s−1 rm dependence and the radial vorticity at the surface is no longer
above 50◦ latitude) is the m = 1 mode with north-south anti- maximum at the equator at azimuthal wavenumbers m & 5. Sec-
symmetric longitudinal velocity vφ with respect to the equator. ondly, we report a new class of modes with frequencies close
This m = 1 mode was identified by Gizon et al. (2021) using to that of the classical Rossby modes. They share properties of
linear calculations in two-dimensional model, which are further both equatorial Rossby modes and convective modes. Thirdly,
discussed in the present paper and Fournier et al. (2022). It cor- we provide a physical explanation for the properties of the m = 1
responds to the spiral-like velocity feature reported at high lat- high-latitude modes in terms of the baroclinic instability due to
itudes by Hathaway et al. (2013), although it was reported as the latitudinal entropy gradient in the convection zone.
giant-cell convection in that work. The organization of the paper is as follows. In Sect. 2 we
The equatorial-Rossby and high-latitude modes involve specify the linearized equations and solve the eigenvalue prob-
mostly toroidal motions with a radial velocity that is small lem. The low-frequency modes are discussed in Sect. 3 for the
compared to the horizontal velocity components. Non-toroidal inviscid, adiabatically stratified, and uniformly rotating case.
inertial modes have also been theoretically studied, mainly Then, the effects of turbulent diffusion and a non-adiabatically
in the context of incompressible fluids. These modes tend to stratified background are discussed in Sects. 4 and 5. We discuss
be localized onto so-called attractors, closed periodic orbits how the solar differential rotation and the associated baroclinic-
of rays reflecting off the spherical boundaries (Maas & Lam ity affect the mode properties in Sect. 6. The results are summa-
1995; Rieutord & Valdettaro 1997, 2018; Rieutord et al. 2001; rized in Sect. 7.
Sibgatullin & Ermanyuk 2019). They are also strongly affected
by critical latitudes when differential rotation is included (e.g.,
Baruteau & Rieutord 2013; Guenel et al. 2016). 2. Eigenvalue problem
In numerical simulations of solar-like rotating convection, In order to investigate the properties of various inertial modes in
equatorial convective columns aligned with the rotation axis the Sun, a new numerical code has been developed. We consider
are prominent (e.g., Miesch et al. 2008; Bessolaz & Brun 2011; the linearized fully-compressible hydrodynamic equations in a
Matilsky et al. 2020). They are known as “Busse columns” (after spherical coordinate (r, θ, φ).
Busse 1970), or “thermal Rossby waves”, or “banana cells”
in the literature. We refer to them as “columnar convective
modes” in the remainder of this paper. These convective columns 2.1. Linearized equations
propagate in the prograde direction owing either to the “topo- The linearized equations of motion, continuity, and energy con-
graphic β-effect” originating from the geometrical curvature servation are:
(e.g., Busse 2002) or to the “compressional β-effect” originating
∂u ∇p1 ρ1 ∂u
from the strong density stratification (Ingersoll & Pollard 1982; = − − ger − (Ω − Ω0 ) − 2Ωez × u
Evonuk 2008; Glatzmaier et al. 2009; Evonuk & Samuel 2012; ∂t ρ0 ρ0 ∂φ
Verhoeven & Stellmach 2014). Glatzmaier & Gilman (1981) 1
− r sin θ u · ∇Ω + ∇ · D, (1)
numerically derived the dispersion relation and the radial eigen- ρ0
functions of these convective modes using a one-dimensional ∂ρ1 ∂ρ1
cylinder model. They showed that the fundamental (n = 0) mode = − ∇ · (ρ0 u) − (Ω − Ω0 ) , (2)
is the fastest of these prograde propagating modes with an eigen- ∂t ∂φ
function that is localized near the surface, where the compres- ∂s1 vr vθ ∂s0 ∂s1
= cp δ − − (Ω − Ω0 )
sional β-effect is strongest. ∂t Hp r ∂θ ∂φ
In the parameter regime of the various numerical simula- 1
tions, the columnar convective modes are the structures that + ∇ · (κρ0 T 0 ∇s1 ), (3)
ρ0 T 0
are the most efficient to transport thermal energy upward under
the rotational constraint (e.g., Gilman 1986; Miesch et al. 2000, where, u = (vr , vθ , vφ ) is the 1st-order velocity perturbation. In
2008; Brun et al. 2004; Käpylä et al. 2011; Gastine et al. 2013; this paper, we only consider the differential rotation for the mean
Hotta et al. 2015; Featherstone & Hindman 2016; Matilsky et al. flow and ignore meridional circulation. Thus, the background
2020; Hindman et al. 2020). Furthermore, it is often argued velocity is U = r sin θ(Ω − Ω0 )eφ . Here, Ω is a function of r
that these convective modes play a critical role in transporting and θ, denoting the rotation rate in the Sun’s convection zone,
the angular momentum equatorward to maintain the differen- and Ω0 is the rotation rate of the observer’s frame. We note that,
tial rotation of the Sun (e.g., Gilman 1986; Miesch et al. 2000; in this work, we start by analyzing the case with uniform rota-
Balbus et al. 2009). The dominant columnar convective modes tion for simplicity. In this case, Ω0 represents the rotation rate
seen in simulations have not been detected in the velocity field of the unperturbed background state. For the case with the solar
at the surface of the Sun. However, we show in this paper that differential rotation, we chose to use the Carrington rotation rate
some retrograde inertial modes have a mixed character and share Ω0 /2π = 456.0 nHz.
some properties with columnar convection. The unperturbed model is given by p0 , ρ0 , T 0 , g, and Hp
In this paper, we study the properties of the equatorial which are the pressure, density, temperature, gravitational accel-
Rossby modes, the high-latitude inertial modes, and the colum- eration, and pressure scale height of the background state. The
nar convective modes in the linear regime. We are mainly background is assumed to be spherically symmetric and in an
interested in the effects of turbulent diffusion, solar differen- adiabatically-stratified hydrostatic balance. All of these variables
tial rotation, and non-adiabatic stratification on these modes. We are functions of r alone. We use the same analytical model as
note that the critical-latitude modes, discussed in Fournier et al. Rempel (2005) and Bekki & Yokoyama (2017) for the back-
(2022), are not dealt with in depth in this paper. ground stratification which nicely mimics the solar model S
A16, page 2 of 23
Y. Bekki et al.: Linear model of solar inertial oscillations
(Christensen-Dalsgaard et al. 1996). The variables with sub- velocity vφ , density perturbation ρ1 , and entropy perturbation s1
script 1, p1 , ρ1 , and s1 , represent the first-order perturbations are out of phase with the meridional components of velocity (vr
of pressure, density, and entropy that are associated with veloc- and vθ ) in the inviscid limit (v = κ = 0).
ity perturbation, u. Here, to close the equations, the linearized Equations (6)–(10) can be combined into an eigenvalue
equation of state is used as follows: problem:
p1 ρ1 s1 ωV = MV, (11)
=γ + , (4)
p0 ρ0 cv
where
where γ = 5/3 is the specific heat ratio and cv denotes the spe-
cific heat at constant volume.
vr
vθ
Although the background is approximated to be adiabatic,
V = vφ ,
we can still introduce a small deviation from the adiabatic strat- (12)
ification in terms of the superadiabaticity δ = ∇ − ∇ad , where
ρ1
∇ = d ln T/d ln p is the double-logarithmic temperature gradi- s1
ent. In the solar convection zone, superadiabaticity is estimated and M is the linear differential operator represented by the right-
as δ ≈ 10−6 (e.g., Ossendrijver 2003). Also, when the solar dif- hand side of the Eqs. (6)–(10). The operator M depends on
ferential rotation is included, we may add a latitudinal entropy azimuthal order m and the model parameters such as differential
variation ∂s0 /∂θ that is associated with the thermal wind bal- rotation, Ω(r, θ), superadiabaticity, δ, and diffusivities, ν and κ.
ance of the differential rotation (e.g., Rempel 2005; Miesch et al.
2006; Brun et al. 2011).
We assume that the viscous stress tensor, D, is given by 2.3. Boundary conditions
2
! In this study, we confine our numerical domain from rmin =
Di j = ρ0 v Si j − δi j ∇ · u , (5) 0.71 R to rmax = 0.985 R in the radial direction to avoid the
3
strong density stratification near the solar surface and gravity
where S is the deformation tensor. We refer to Fan & Fang modes in the radiative interior. Because of viscosity, in this prob-
(2014) (their Eqs. (8)–(13)) for more detailed expressions of Si j lem we have four second-order (in both the radial and latitudinal
in spherical coordinates. The viscous and thermal diffusivities directions) PDEs and one first-order PDE. Equation (9) does not
are denoted by v and κ respectively. increase the order of the system as ρ1 can be eliminated from the
system without increasing the order of the other equations. Thus,
eight boundary conditions are required in the radial direction
2.2. Eigenvalue problem (four at the top, four at the bottom). At the top and bottom, we
We assume that the φ and t dependence of all the perturbations u, use impenetrable horizontal stress-free conditions for the veloc-
ρ1 , p1 , and s1 is given by the waveform exp [i(mφ − ωt)], where ity and assume there is no entropy flux (∝κ∂s1 /∂r) across the
m is the azimuthal order (an integer) and ω is the complex angu- boundary:
lar frequency. With this representation, Eqs. (1)–(3) give ∂ vθ ∂ vφ ∂s1
vr = 0, = = 0, = 0. (13)
∂ ρ1 s1 g ∂r r ∂r r ∂r
" !#
ωvr = − i Cs2 + + i s1 + 2iΩ sin θvφ
∂r ρ0 cp cp All latitudes are covered in the numerical scheme, from the north
i pole (θ = 0) to the south pole (θ = π). We need another eight
+ m(Ω − Ω0 )vr + (∇ · D)r , (6) boundary conditions in the θ direction. For non-axisymmetric
ρ0
cases (m , 0), at the poles we impose
i ∂ ρ1 s1
" !#
ωvθ = − Cs2 + + 2iΩ cos θ vφ vr = vθ = vφ = 0, s1 = 0, (14)
r ∂θ ρ0 cp
i to make the quantities single valued. For the axisymmetric case
+ m(Ω − Ω0 )vθ + (∇ · D)θ , (7)
ρ0 (m = 0), at both poles we assume instead
mCs2 ρ1 s1 ∂ vφ
!
ωvφ = − + − 2iΩ(vr sin θ + vθ cos θ) ∂vr ∂s1
r sin θ ρ0 cp = vθ = 0, = 0, = 0. (15)
∂θ ∂θ sin θ ∂θ
∂Ω vθ ∂Ω
!
+ m(Ω − Ω0 )vφ − ir sin θ vr +
∂r r ∂θ 2.4. Numerical scheme
i We numerically solve the above eigenvalue problem using a
+ (∇ · D)φ , (8)
ρ0 finite differencing method in the meridional plane. We use a
ρ0 spatially-uniform grids. The grids for vφ , ρ1 , and s1 are staggered
ωρ1 = − iρ0 ∇ · u + i vr + m(Ω − Ω0 )ρ1 , (9) grids by half a grid point in radius for vr and half a grid point
Hρ
cp δ in colatitude for vθ (following Gilman 1975), as is illustrated in
i ∂s0
ωs1 = i vr − vθ + m(Ω − Ω0 )s1 Fig. 1. Spatial derivatives are evaluated with a centered second-
Hp r ∂θ order accurate scheme. By converting the two dimensional grid
i (Nr , Nθ ) into one dimensional array with the size Nr , Nθ for all
− ∇ · (κρ0 T 0 ∇s1 ) , (10)
ρ0 T 0 variables, V is defined as a one dimensional vector with size
∼5Nr Nθ . Once the boundary conditions are properly set, M can
where Cs = (γp0 /ρ0 )1/2 is the sound speed and cp = γcv is the be constructed as a two-dimensional complex matrix with the
constant specific heat at constant pressure. Here, the longitudinal size approximately (5Nr Nθ × 5Nr Nθ ). This method is similar to
A16, page 3 of 23A&A 662, A16 (2022)
that of Guenther & Gilman (1985). In practice, each element of
M can be computed by substituting a corresponding unit vector
V into the right-hand side of the Eqs. (6)–(10). In most of the
calculations, we use the grid resolution of (Nr , Nθ ) = (16, 72).
We have also carried out higher-resolution calculations with
(Nr , Nθ ) = (24, 180) for a uniform rotation case to check the grid
convergence of the results. When the grid resolution is increased,
the total number of eigenmodes increases accordingly. The addi-
tional modes have higher radial and latitudinal wavenumbers and
are more finely structured. For the interpretation of the large-
scale modes which have been observed on the Sun, the results
are converged with (Nr , Nθ ) = (16, 72).
We use the LAPACK routines (Anderson et al. 1999) to
numerically compute the eigenvalues and eigenvectors of
M(m, ν, κ, δ, Ω), corresponding to the mode frequencies, ω, and
the eigenfunctions (vr , vθ , vφ , ρ1 , s1 ) of linear modes in the Sun.
In this study, we limit the range of azimuthal orders to m ≥ 0
and allow the real frequency to take a negative value. This means
that 0 corresponds to exponentially growing modes.
2.5. Example spectrum for uniform rotation
Fig. 1. Layout of the staggered grid used to solve the eigenvalue equa-
For each m, there are 5Nr Nθ eigensolutions with frequencies ω tion. The grid locations where vφ , ρ1 , and s1 are defined are denoted by
and eigenfunctions V. As an example, we show the typical dis- red circles. The blue and green circles represent the grid locations of vr
tribution of the output eigenfrequencies in a complex plane for and vθ , respectively. The grid resolution is reduced for a visualization
the case with m = 1, δ = 10−6 (weakly superadiabatic) and purpose.
ν = κ = 2 × 1012 cm2 s−1 in Fig. 2. We note that the differential
rotation is not included here for simplicity; the uniform rotation
rate Ω is equal to the Carrington rotation rate Ω0 .
The modes belong to one of several regions in the complex
eigenfrequency spectrum. The modes seen in Fig. 2a are acous- ences and, then, the effects of turbulent diffusion, non-adiabatic
tic modes (p modes) that are slightly damped due to the vis- stratification, and differential rotation are compared to these ref-
cous and thermal diffusion. On this plot, the effect of rotation erence results.
is not visible to the eye. In the rest of this paper, we focus on In the inviscid case with uniform rotation, M is self adjoint,
the low-frequency modes in the inertial frequency range. Iner- thus the physically-meaningful solutions must have real eigen-
tial oscillations are confined within the range | 0), we can see that some of vr and vθ have the same complex phase on each meridional
modes have positive imaginary frequencies (=[ω] > 0) at very plane, and those of vφ , ρ1 are 90◦ out of phase with respect to
low frequencies and thus are unstable. These convective modes vr and vθ . In presenting the results in this section, we choose a
are shown in Fig. 2c. meridional plane where vr and vθ are real.
When the background is weakly subadiabatic (e.g., δ = In the following sections, we conduct a mode-by-mode anal-
−10−6 ), all the modes become stable (=[ω] < 0) and some iner- ysis for the equatorial Rossby modes with no radial nodes (n =
tial modes are partially mixed with gravity modes (g modes). 0) and one radial node (n = 1), columnar convective modes (ther-
When Ω0 = 0, the modes are either purely convective modes or mal Rossby waves) with both north-south symmetries, and the
purely g modes depending on the sign of δ as shown in Fig. 3. high-latitude modes with both north-south symmetries. Funda-
The frequency of the g modes depends on δ and, depending on mental properties of these modes are summarized in Table 1.
Ω0 , can lie in the inertial range. Their dispersion relations are presented in Table 2.
3.1. Equatorial Rossby modes
3. Reference case: No diffusion, adiabatic
In this section, we discuss the equatorial Rossby modes (r
stratification, uniform rotation modes). The modes with no radial nodes (n = 0) and one radial
In this section, we report the results of an ideal case where turbu- node (n = 1) are reported.
lent viscous and thermal diffusivities are set to zero (ν = κ = 0),
the background is convectively neutral (δ = 0), and no differ- 3.1.1. n = 0 modes
ential rotation is included (Ω(r, θ) = Ω0 and ∂s0 /∂θ = 0). We
present the dispersion relations and eigenfunctions of various In order to extract the n = 0 equatorial Rossby mode at each m,
types of global-scale vorticity modes that might be relevant to we apply the following procedure to the computed eigenfunc-
the Sun. We go on to use the results of this ideal setup as refer- tions V. The latitudinal and longitudinal velocities at the surface
A16, page 4 of 23Y. Bekki et al.: Linear model of solar inertial oscillations
Fig. 2. Overview of our linear eigenmode calculation in the case of uniform rotation (Ω = Ω0 ), a weakly superadiabatic stratification (δ =
10−6 ), and moderate turbulent viscous and thermal diffusivities (ν = κ = 1011 cm2 s−1 ). Shown are the results at m = 8. Upper panels: complex
eigenfrequencies ω in the co-rotating frame. Panel a: real frequencies in the range ±10 mHz showing the acoustic modes (p modes). Panel b:
zoom-in focusing on the inertial range | 0). Lower panels:
example eigenfunctions of pressure (acoustic), non-toroidal inertial, toroidal inertial (equatorial Rossby), and columnar convective modes, from
left to right. The eigenfrequencies of these modes are highlighted by orange, green, blue, and red dots in the upper panels.
are projected onto a basis of associated Legendre polynomials: eigenfunctions for m = 2 and 8, respectively. The eigenfunctions
are normalized such that the maximum of vθ is 2 m s−1 at the
lmax
X surface. The amplitude of radial velocity vr is about 103 times
vθ (rmax , θ) = l (cos θ),
al−m Pm (16) smaller than those of horizontal velocities, vθ and vφ , implying
l=0 that the fluid motion is essentially toroidal. We find that using a
lmax
X higher resolution leads to even smaller values of vr . The pressure
vφ (rmax , θ) = l (cos θ),
bl−m Pm (17) perturbation, p1 , is positive (negative) where the radial vorticity,
l=0 ζr , is negative (positive) in the northern (southern) hemisphere,
where lmax = 2Nθ /3 − 1 = 47. We also compute the number which is consistent with the modes being in geostrophic balance.
of radial nodes, n, of vθ at the equator. We select the modes that As m increases, the n = 0 equatorial Rossby modes are shifted
satisfy all of the following three criteria: The l = m component of to the surface and to the equator. The horizontal eigenfunction
vθ is dominant (|a0 | > |a j | for all j > 0); the l = m + 1 component of ζr becomes more elongated in latitude, which means that vθ
of vφ is dominant (|b1 | > |b j | for all j , 1); and the number of becomes much stronger than vφ to keep the mass conservation
radial nodes of vθ is zero at the equator, n = 0. horizontally.
Figure 4a shows the dispersion relation of the selected n = 0 Figure 5a shows the radial structure of the eigenfunctions
equatorial Rossby modes for this ideal setup for m = 1−16. of vθ at the equator for selected azimuthal orders m. Solid and
It should be noted that these modes are the only type of iner- dashed lines compare our results with the analytical solution
tial modes where a simple analytical solution can be found in vθ ∝ rm . It is seen that computed eigenfunctions exhibit the
the inviscid, uniformly-rotating limit (e.g., Saio 1982). There- rm dependence that agree with the analytical solutions. We also
fore, we use this analytical solution to verify our code. The red confirm the same rm dependence for the eigenfunctions of vθ in
points and black dashed lines represent the computed eigen- the middle latitudes (not shown). For higher m, the radial eigen-
frequencies in our model and the theoretically-expected dis- function shows a slight deviation (within a few percent error)
persion relation, ω = −2Ω0 /(m + 1), respectively. We find from the analytical solution. This is possibly due to the stress-
that the differences in the normalized frequencies are less than free boundary condition, ∂(vθ /r)/∂r = 0, at the top and bottom
10−2 at all m. boundaries, which conflicts with the rm dependence. Figure 5b
The typical flow structure of this mode is schematically illus- shows the latitudinal eigenfunctions of vθ at the surface. Again,
trated in Fig. 4b where the volume rendering of the radial vortic- an agreement can be seen between our results and the analytical
ity ζr is shown by red and blue. Figures 4c and d show the real solutions vθ ∝ sinm−1 θ.
A16, page 5 of 23A&A 662, A16 (2022)
inertial inertial
& &
convective g-modes
pure pure
convective g-modes
Fig. 3. Eigenfrequency spectrum in the complex plane at m = 8 for (a) δ = 10−6 , Ω = Ω0 , (b) δ = −10−6 , Ω = Ω0 , (c) δ = 10−6 , Ω = 0, and (d)
δ = −10−6 , Ω = 0, respectively. Here, Ω0 is the Carrington rotation rate. Only inertial frequency range is shown. Upper and lower panels: cases
with and without uniform rotation. Left and right panels: cases with superadiabatic and subadiabatic background. Panel a: is the same as Fig. 2b.
Table 1. Summary of the properties of the modes of the models discussed in this paper.
Classification Peak location North-south Propagation Sections
of kinetic energy symmetries direction discussed
vr vθ vφ
Equator A S A Retrograde Sects. 3.1.1, 4, 6.1
Equator A S A Retrograde Sects. 3.1.2, 6.1
Equator S A S Prograde Sects. 3.2.1, 5
Equator A S A Prograde Sect. 3.2.2
Near poles S A S Retrograde Sect. 3.3.1
Near poles A S A Retrograde Sects. 3.3.2, 6.2
Notes. Each row refers to a set of modes with different m values. The integer n denotes the number of radial nodes of vθ at the equator for the Rossby
modes. The north-south symmetries of the different components of the velocity are given in Cols. 3–6, where “S” indicates the velocity component
is symmetric across the equator and “A” indicates the velocity component is antisymmetric across the equator. The propagation direction is for the
uniformly rotating case, and is given in the rotating frame.
3.1.2. n = 1 modes Figure 6a shows the dispersion relation of the selected n = 1
The equatorial Rossby modes with one radial node (n = 1) can equatorial Rossby modes for 0 ≤ m ≤ 16. It should be noted
be selected by applying the following filters for latitudinal and that we successfully identify the axisymmetric mode (m = 0) at
longitudinal velocity eigenfunctions: we select the modes whereY. Bekki et al.: Linear model of solar inertial oscillations
Table 2. Dispersion relations of the modes of the model with uniform rotation (Ω = Ω0 ), ν = κ = 0, and δ = 0.
mA&A 662, A16 (2022)
(a) (b)
1.0 this study 1.0 this study
rm sinm 1
0.8 0.8
v at equator
v at surface
0.6 m=2 0.6
0.4 m=4 0.4
0.2 m=8 0.2
m = 16
0.0 0.0
0.75 0.80 0.85 0.90 0.95 75 50 25 0 25 50 75
r/R Latitude [deg]
Fig. 5. (a) Radial structure of the eigenfunction of vθ at the equator for the n = 0 equatorial Rossby modes in the inviscid, uniformly rotating,
and adiabatically stratified case. Overplotted dashed lines represent theoretically predicted radial dependence vθ ∝ rm . The eigenfunctions are
normalized to unity at the surface r = rmax . (b) Latitudinal structure of the eigenfunction of vθ at the surface. Dashed lines are the theoretical
solution in the form of Legendre polynomials, vθ ∝ sinm−1 θ. All the eigenfunctions are normalized at the equator.
Fig. 6. Dispersion relation and eigenfunctions of the equatorial Rossby modes with one radial node (n = 1) in the inviscid, uniformly rotating,
adiabatically stratified case. The same notation as Fig. 4 is used here.
direction with slower phase speed than that of n = 0 Rossby Figure 7a shows the radial structure of the eigenfunctions of
modes at low m. However, for m ≥ 8, the mode frequencies vθ at the equator for selected m. It is clearly seen that the loca-
become so close to those of n = 0 modes that they are almost tion of the radial node shifts towards the surface as m increases.
indistinguishable. Figure 7b shows the latitudinal structure of the eigenfunctions
Figure 6b shows a schematic sketch of typical flow motion of vθ at the surface. The eigenfunctions peak at the equator and
of the n = 1 equatorial Rossby mode. Figures 6c and d further change their sign in the middle latitudes (25◦ −50◦ ) and decay at
shows the obtained eigenfunctions of n = 1 equatorial Rossby higher latitudes.
modes plotted in the same way as in Fig. 4. It is clearly shown
that vθ has a nodal plane in the middle convection zone at the 3.2. Columnar convective modes
equator which extends in the direction of the rotation axis. One
of the most striking consequences of the existence of the radial In this section, we carry out a similar mode-by-mode analy-
node is that substantial vr is involved, owing to the radial shear sis for the columnar convective modes (thermal Rossby waves)
of vθ . Therefore, unlike the n = 0 modes, the associated fluid with both hemispheric symmetries. Here, we define the north-
motions are no longer purely toroidal and become essentially south symmetry based on the eigenfunction of z-vorticity ζz . The
three-dimensional. “banana cell” convection pattern can be essentially regarded as
A16, page 8 of 23Y. Bekki et al.: Linear model of solar inertial oscillations
(a) (b)
1.0 m=2 1.0
m=6
0.8 m = 10 0.8
m = 16
0.6 0.6
v at equator
v at surface
0.4 0.4
0.2
0.2
0.0
0.0
0.2
0.2
0.75 0.80 0.85 0.90 0.95 75 50 25 0 25 50 75
r/R Latitude [deg]
Fig. 7. (a) Radial structure of the eigenfunction of vθ at the equator of the n = 1 equatorial Rossby modes in the inviscid, uniformly rotating,
adiabatically stratified case. The eigenfunctions are normalized to unity at the surface r = rmax . (b) Latitudinal structure of the eigenfunction of vθ
at the surface normalized at the equator.
Fig. 8. Dispersion relation and eigenfunctions of the north-south ζz -symmetric columnar convective modes in the case of uniform rotation, no vis-
cosity, and adiabatic stratification. (a) Dispersion relation of the north-south ζz -symmetric columnar convective modes in red points. For compari-
son, dispersion relation analytically derived using one-dimensional cylinder model by Glatzmaier & Gilman (1981) is overplotted in black dashed
line. (b) Schematic illustration of flow structure of the mode. Red and blue volume rendering shows the structure ofA&A 662, A16 (2022)
Fig. 9. Dispersion relation and eigenfunctions of the north-south ζz -antisymmetric columnar convective modes in the inviscid, uniformly rotating,
adiabatically stratified case. The same notation as Fig. 8 is used. Panel a: the dispersion relation of the north-south ζz -symmetric columnar
convective modes is shown in black dashed line for comparison.
Glatzmaier & Gilman (1981) (their Fig. 2). Qualitatively, they
Rossby (n = 1)
both show similar features: columnar convective modes prop- 1.0 Convective
agate in a prograde direction at all m. The modes are almost ( z antisym)
non-dispersive at low m (≤7), but at higher m, the mode fre-
quencies become almost constant atY. Bekki et al.: Linear model of solar inertial oscillations
Fig. 11. Dispersion relation and eigenfunctions of the high-latitude modes with north-south symmetric ζz in the inviscid, uniformly rotating,
adiabatically stratified case. The same notation as Fig. 8 is used. Panel a: the dispersion relation of the l = m + 1 Rossby modes is shown in black
dashed line.
Fig. 12. Dispersion relation and eigenfunctions of north-south ζz -antisymmetric high-latitude modes in the inviscid, uniformly rotating, adiabati-
cally stratified case. The same notation as Fig. 11 is used. Panel a: the dispersion relation of the l = m + 2 Rossby mode is shown in black dashed
line.
faster phase speed than that of the ζz -symmetric modes. At high are antisymmetric across the equator. It should be noted that
m, the dispersion relation asymptotically approaches that of the strong latitudinal motions are involved at the equator at the
ζz -symmetric modes. surface.
Figures 9c and d show the example eigenfunctions of the We find that the eigenfunctions of the m = 0 mode are
north-south ζz -antisymmetric columnar convective modes. The the complex conjugate of the n = 1 equatorial Rossby mode,
flow structure is dominantly characterized by z-vortex tubes that which means that these two modes are identical at m = 0
A16, page 11 of 23A&A 662, A16 (2022)
(we note the phase speed no longer matters in the case of the
non-propagating axisymmetric mode). To better illustrate this
point, we show in Fig. 10 the dispersion relations of these
two modes in the full (m, 0.5, where Ein and ECZ are the volume-integrated
kinetic energies inside the tangent cylinder and in the entire con- So far, we have discussed the results for an inviscid case. In this
vection zone, respectively; the l = m + 1 component of vθ is section, we examine the effects of viscous and thermal diffusion
dominant at the bottom of the convection zone; and the number arising from turbulent mixing of momentum and entropy in the
of z-nodes of vθ is zero at $ = 0.5 R . Sun (e.g., Rüdiger 1989). We start our discussion by estimat-
Figure 11a shows the dispersion relation of the north-south ing the impact of the turbulent diffusion on (classical) Rossby
ζz -symmetric high-latitude modes. We find the high-latitude modes. The oscillation period of the equatorial Rossby mode at
modes are much more dispersive than the columnar convec- the azimuthal order m is given by
tive modes at low m. The dispersion relation is found to be
2π 2Ω0
roughly approximated by the non-sectoral Rossby modes’ dis- PRo = , where ωRo = − · (18)
persion relation with one latitudinal node (l = m+1), as shown in ωRo m+1
the black dashed line in Fig. 11a. This is because the horizontal
On the other hand, typical diffusive timescale can be estimated
flows at the bottom boundary behave like the l = m+1 (classical)
as:
Rossby modes. We note, however, that this is not regarded as the
mode mixing as discussed in Sect. 3.2.2. 2
lm R
Figures 11c and d show example eigenfunctions of the ζz - τdiff = , with lm = , (19)
ν m
symmetric high-latitude modes. The fluid motion is predomi-
nantly characterized by z-vortices inside the tangential cylinder where lm denotes the typical length scale of the Rossby mode.
in both hemispheres, as schematically illustrated in Fig. 11b. Figure 13 compares PRo and τdiff as functions of m. Two repre-
The power of ζz peaks at the tangential cylinder $ = rmin . We sentative values of turbulent diffusitivies in the solar convection
note that the longitudinal velocity vφ extends slightly outside the zone ν = 1012 and 1013 cm2 s−1 are shown (e.g., Ossendrijver
cylinder. Again, =[p1 ]=[ζz ] < 0 follows from the mode being in 2003). When PRo τdiff , viscous diffusion is almost negligible.
geostrophic balance. However, if PRo & τdiff , diffusion can have a dominant effect on
the Rossby modes. For a given turbulent diffusivity ν, the critical
azimuthal order, mcrit , can be defined as:
3.3.2. North-south ζz -antisymmetric modes
!1/3
North-south ζz -antisymmetric high-latitude modes are selected R Ω0
mcrit = . (20)
using the following filters: The kinetic energy is predominantly πν
A16, page 12 of 23Y. Bekki et al.: Linear model of solar inertial oscillations
Fig. 14. Eigenfrequency spectra of the low-frequency vorticity modes in a complex plane with different values of diffusivities for (a) m = 2 and
(b) m = 16. Different colors represent different classes of inertial modes. Different symbols represent different values of the viscous and thermal
diffusivities. In all cases, rotation is uniform and the stratification is adiabatic.
Fig. 15. e-folding lifetimes of various low-frequency modes for a viscous diffusivity (a) ν = 1011 cm2 s−1 and (b) ν = 1012 cm2 s−1 . We note that
all the modes selected here are stable modes (=[ω] < 0). Different colors represent different types of inertial modes. The horizontal black dashed
line shows the length of the SDO/HMI observational record (T obs ≈ 12 yr as of today). In both cases, rotation is uniform and the stratification is
adiabatic. The lifetimes of the convective modes and high-latitude modes are very sensitive to the radial and latitudinal entropy gradients, a point
that is discussed in Sects. 5 and 6.2.
The Rossby modes are dominated by diffusive effects for m > similar degree. We note that a strong diffusion modifies not only
mcrit . Figure 13 implies that the Rossby modes in the Sun are the imaginary part but also the real part of the mode frequen-
substantially affected by the turbulent diffusion, especially for cies. The computed e-folding times of these modes |=[ω]|−1 are
m ≥ 5−6. shown in Fig. 15 for the two representative values of turbulent
In this paper, we carry out a set of calculations of viscosity ν = 1011 and 1012 cm2 s−1 .
uniformly-rotating adiabatic fluid with varying diffusivities; ν = We go on to focus on the n = 0 equatorial Rossby modes
109 , 1010 , 1011 , 1012 , and 1013 cm2 s−1 . For simplicity, we fixed to see how eigenfunctions are affected by the viscous diffu-
the Prandtl number to unity so that κ = ν. In this case, both sion. Figure 16a shows the real (top row) and imaginary (bottom
the eigenfrequencies and eigenfunctions are complex. Figure 14 row) eigenfunctions of radial vorticity ζr at m = 16 for differ-
shows the eigenfrequencies of the six types of Rossby modes ent values of viscous diffusivities ν. As ν increases, the n = 0
discussed in Sect. 3 for different viscous diffusivities in a com- equatorial Rossby modes are shifted towards the base of the
plex plane. Figures 14a and b show the cases for m = 2 and 16, convection zone. This is clearly illustrated in Fig. 16b, where
respectively. In general, the modes are damped by diffusion so the absolute amplitudes of radial vorticity at the equator are
that the imaginary frequencies are shifted towards more negative shown as functions of radius. When ν becomes sufficiently large,
values. At small m (e.g., m = 2), diffusion tends to act predomi- the radial eigenfunction substantially deviates from the well-
nantly on the columnar convective modes with both symmetries known rm dependence. This can be explained as follows: with the
and n = 1 equatorial Rossby modes, whereas the n = 0 Rossby moderate diffusion included, the radial force balance between
modes and the high-latitude modes remain almost unaffected. At Coriolis force and pressure gradient force is no longer main-
large m (e.g., m = 16), however, all the modes are damped to a tained. Consequently, radial flows are driven and the diffusive
A16, page 13 of 23A&A 662, A16 (2022)
Fig. 16. (a) Meridional eigenfunctions of radial vorticity ζr of the n = 0 equatorial Rossby mode at m = 16 for different values of viscous
diffusivities ν. Upper and lower panels: normalized real and imaginary eigenfunctions, respectively. (b) Radial eigenfunctions of |ζr | at the equator
(normalized by their maximum amplitudes). Different colors represent different values of diffusivities. In all cases, rotation is uniform and the
stratification is adiabatic.
16
16
subadiabatic
5
4
6
5
superadiabatic
2 3 4 5 6 7 8
1 3 6 7 8
4 5 16 16
2
3 16
3 4
Fig. 17. (a) Dispersion relations of the north-south ζz -symmetric columnar convective modes with different background superadiabaticity values
δ. Different colors represent different values of superadiabaticity. Circles and diamonds denote the stable (=[ω] < 0) and unstable (=[ω] > 0)
modes, respectively. (b) Eigenfrequencies in the complex plane. Each circle (diamond) represent a mode with azimuthal order m, which is labeled
with small integers from m = 1 to 16. In all cases, rotation is uniform and the diffusivities are set ν = κ = 1012 cm2 s−1 .
momentum flux becomes directed radially inward. In fact, the vary the superadiabaticity from weakly subadiabatic to weakly
confinement of the n = 0 equatorial Rossby modes near the superadiabatic, δ = −2 × 10−6 , −10−6 , 0, 10−6 , 2 × 10−6 , while
base is also seen in rotating convection simulations where the keeping the diffusivities fixed (ν = κ = 1012 cm2 s−1 ). The
diffusion can be significantly enhanced by turbulent convection solar differential rotation and latitudinal entropy gradient are
(Bekki et al., in prep.). not included. Since the entropy perturbation is generated by
the radial flow, in this section, we focus on the (north-south
ζz -symmetric) columnar convective modes where strong radial
5. Effect of non-adiabatic stratification motions are involved.
In this section, the effects of non-adiabatic stratification are Figure 17a shows the dispersion relations of the ζz -
investigated. While theoretical models of the Sun conventionally symmetric columnar convective modes for different δ. As the
assume a slightly positive superadiabaticity value 0 < δ . 10−6 background becomes more subadiabatic (superadiabatic), the
(e.g., Ossendrijver 2003), recent numerical simulations of solar mode frequencies become higher (lower), that is, the modes
convection imply that the lower half of the convection zone propagate in a prograde direction with faster (slower) phase
might be slightly subadiabatic (Hotta 2017; Käpylä et al. 2017, speed. When δ is sufficiently large, the imaginary mode frequen-
2019; Bekki et al. 2017; Karak et al. 2018). To this end, we cies become positive, namely, the modes become convectively
A16, page 14 of 23Y. Bekki et al.: Linear model of solar inertial oscillations Fig. 18. Radial velocity vr (upper plots) and entropy perturbation s1 (lower plots) of the columnar convective modes along the rotational axis displayed in the equatorial plane for subadiabatic and superadiabatic background. Panels a and b: cases with subadiabatic background δ = −2×10−6 for m = 3 and m = 8, respectively. Panels c and d: are the same plots for a superadiabatic background δ = 2 × 10−6 . The eigenfunctions are normalized such that the maximum radial velocity is 10 m s−1 at the equator. In all cases, rotation is uniform and the diffusivities are set ν = κ = 1012 cm2 s−1 . Fig. 19. Transport properties of thermal energy and angular momen- tum by the north-south ζz -symmetric columnar convective modes for Fig. 20. Solar differential rotation profile used in this study. (a) Differ- m = 16. (a) Correlation between radial velocity velocity and tempera- ential rotation Ω(r, θ) in a meridional plane, deduced from the global ture perturbation hvr T 1 i, (b) Reynolds stress between radial and longi- helioseismology (Larson & Schou 2018). (b) Latitudinal profiles of dif- tudinal velocities hvr vφ i, and (c) Reynolds stress between latitudinal and ferential rotation at different depths. Horizontal dashed lines indicate longitudinal velocities hvθ vφ i. The background is weakly superadiabatic the theoretically-expected phase speed of the sectoral (l = m) clas- (δ = 2 × 1016 ), rotation is uniform, and moderate diffusivities are used sical Rossby modes for selected azimuthal orders m = 2, 3, 4, 8, 16. (ν = κ = 1012 cm2 s−1 ). The eigenfunctions are normalized such that the The observing frame is chosen to be the Carrington frame rotating at maximum radial velocity is 10 m s−1 at the equator. Ω0 /2π = 456.0 nHz. unstable. This is clearly manifested in Fig. 17b where the mode quencies become higher for δ < 0. The opposite situation hap- frequencies are plotted in a complex plane. Each point denotes pens for δ > 0. Figures 18c and d show the same equatorial cuts each mode with the associated azimuthal order labeled nearby. of vr and s1 for a weakly superadiabatic background. When m is The stable and unstable modes are distinguished by circles and not large enough for the convective instability to occur, it is seen diamonds, respectively. For δ > 0 (blue and purple), a sudden that the phase with positive s1 is behind the phase with positive transition occurs from stable to unstable branches (at m = 5 vr in longitude, leading to a positive correlation between 0. considering whether the buoyancy force acts as a restoring force This effect was first studied in Gilman (1987) using a simplified or the opposite. Figures 18a and b present the snapshots of vr cylindrical model. Figure 18d shows the case where m is suffi- and s1 in an equatorial plane seen from the north pole for a ciently large and the mode becomes convectively unstable. It is weakly subadiabatic background (δ = −2 × 10−6 ) for m = 3 obviously seen that the phases of vr and s1 now coincide and they and 8, respectively. It is seen that the phase with positive s1 is both have the same sign at each phase, leading to hvr s1 i > 0. always ahead of the phase with positive vr in longitude, leading Figure 19 further shows the transport properties of thermal to a negative correlation between
A&A 662, A16 (2022)
(a) m = 2 (b) m = 16
200 200
0 0
200 200
[ ]/2 [nHz]
400 400
600 600
800 800
1000 1000
1200 1200
1000 500 0 500 1000 2000 1000 0 1000
[ ]/2 [nHz] [ ]/2 [nHz]
Fig. 21. Eigenfrequencies of inertial modes under the solar differential rotation in the complex plane for (a) m = 2 and (b) m = 16. The diffusivity
is set to ν = 1012 cm2 s−1 and the background is assumed to be adiabatic, δ = 0. The shaded areas indicate the frequency range associated with the
surface differential rotation, namely, m(Ωpole − Ω0 ) < 2, there emerge critical latitudes where the phase
speed of the Rossby mode matches with the differential rotation
6. Effect of solar differential rotation speed. As discussed in Gizon et al. (2020a) and Fournier et al.
(2022), turbulent viscous diffusion is required to get rid of the
Finally, in this section, we take into account the effects of solar
singularities at the critical latitudes, leading to a formation of vis-
differential rotation. For prescribing Ω(r, θ), we use the data
cous critical layers with the typical latitudinal extent δcrit given
obtained from global helioseismology inversions from MDI and
by
HMI (Larson & Schou 2018) as shown in Fig. 20a. We note that
the observational data is truncated at r = rmin and rmax , and there- !1/3
fore the effects of strong radial shear layers such as tachocline δcrit ν
≈ . (21)
and the near surface shear layer of the Sun are not included. The R mΩ0 R2
A16, page 16 of 23Y. Bekki et al.: Linear model of solar inertial oscillations
Fig. 23. Eigenfunctions of the equatorial Rossby modes with no radial nodes (n = 0). (a) Real (upper) and imaginary (lower) eigenfunctions of
three components of velocity shown in a meridional plane for m = 5. The eigenfunctions are normalized such that the maximum latitudinal velocity
is 2 m s−1 at the surface. The solid black line indicates the location of the critical latitudes where the phase speed of a Rossby mode matches to the
differential rotation sped. (b) Horizontal eigenfunctions of latitudinal velocity vθ (upper) and radial vorticity ζr (lower) at the surface r = 0.985 R
for m = 5. The horizontal black dashed lines indicate the location of the critical latitudes at the surface. (c) and (d) are the counterparts of the
panels a and b for m = 12.
Figure 21 shows the distribution of eigenfrequencies of the perfectly adiabatic and the latitudinal entropy variation ∂s0 /∂θ
global-scale inertial modes in a complex plane for m = 2 and is switched off.
16. Shown in shaded area represent the range of mode fre- Figure 22a shows the dispersion relation of the equatorial
quencies where differential rotation can have a strong impact Rossby modes with n = 0 (red) and n = 1 (blue) for weak
by producing the critical layers. At higher values of m, the (dashed) and strong (solid) viscous diffusivities, respectively.
number of eigenmodes that are affected by differential rota- Shown in white circles, squares, and diamonds are the frequen-
tion increases: In fact, most of the retrograde-propagating cies of the Rossby modes observed on the Sun (Löptien et al.
inertial modes are affected by critical latitudes at higher m 2018; Liang et al. 2019; Proxauf et al. 2020). The viscous dif-
(see Fig. 21b). fusivity value is found to have a rather small effect on the real
part of their eigenfrequencies. At m = 3, the observed fre-
quency agrees almost perfectly with the n = 0 equatorial Rossby
6.1. Rossby modes with viscous critical layers
mode’s frequency. However, for m > 3, the observed frequen-
In this section, we carry out a set of calculations for ν = 1011 and cies lie in between the frequencies of n = 0 and n = 1 modes.
1012 cm2 s−1 with the differential rotation included to study how Figure 22b shows the computed eigenfrequencies in a complex
the equatorial Rossby modes are affected by the viscous critical plane. Unlike the n = 1 modes, the n = 0 modes are substan-
layers. For the sake of simplicity, the background is set to be tially damped only for m ≥ 4, which is likely owing to the
A16, page 17 of 23A&A 662, A16 (2022)
Fig. 24. Eigenfunctions of the equatorial Rossby modes with one radial node (n = 1). Same figure notation as in Fig. 23 is used.
emergence of the critical latitudes that significantly modify the of vθ (and ζr ) peak at the surface and at the equator. Although the
n = 0 modes’ eigenfunctions. The linewidths of the equatorial critical layers exist similarly to the n = 0 modes, they are found
Rossby modes in our model are within the same order of magni- to have a rather limited impact on the n = 1 Rossby modes.
tude as the observations as shown in Fig. 22b. To see the diffusivity dependence, we show ζr at the sur-
Figure 23 shows the velocity eigenfunctions of the n = 0 face for weak (top rows) and strong viscous diffusivities (bot-
modes for the case with ν = 1012 cm2 s−1 . Figures 23a and c tom rows) in Fig. 25. The left and right panels are for the n = 0
show meridional cuts through the eigenfunctions for m = 5 and and n = 1 equatorial Rossby modes, respectively. The solid and
12, respectively. As already discussed in Sect. 4, the latitudinal dashed lines denote the real and imaginary parts, and the ver-
velocity is confined close to the base of the convection zone. tical red line indicates the location of the critical latitudes. The
With differential rotation included, they are further trapped in the phase is defined such thatY. Bekki et al.: Linear model of solar inertial oscillations
(a) ! = 0 mode, $ = 10!! cm2 s-1 (b) ! = 1 mode, $ = 10!! cm2 s-1
(c) ! = 0 mode, $ = 10!" cm2 s-1 (d) ! = 1 mode, $ = 10!" cm2 s-1
Fig. 25. Radial vorticity ζr eigenfunctions at the surface (r = 0.985 R ) for m = 8. Left and right panels: cases for the equatorial Rossby modes
with no radial nodes (n = 0) and with one radial node (n = 1). Upper and lower panels: cases with weak diffusion (ν = 1011 cm2 s−1 ) and
strong diffusion (ν = 1012 cm2 s−1 ). Black solid and dashed lines represent real and imaginary eigenfunctions, respectively. The real part of the
eigenfunctions are defined to be zero at the equator. The vertical red lines denote the location of critical latitudes where the phase speed of a
Rossby mode is equal to the differential rotation velocity.
at m = 8. The Reynolds stresses become substantially non- defined by
zero near the viscous critical layers. It is striking that even
n = 0 mode, which in the case of uniform rotation is toroidal FRS = ρ0 r sin θ hvφ um i, (23)
and non-convective, can transport the angular momentum radi- FMC = ρ0 (r sin θ) Ω um ,
2
(24)
ally upward around the viscous critical layers. Latitudinally, the
angular momentum is transported equatorward in both hemi- FVD = −ρ0 ν(r sin θ) ∇Ω, 2
(25)
spheres. Figure 26c shows the hvθ vθ i at the surface for all m. It is
seen that the correlations become small as m increases because where um is the meridional flow. Figure 27 shows the each
the n = 0 modes are more and more confined closer to the base term of the latitudinal component of the Eq. (22) averaged over
of the convection zone. The counterparts for n = 1 modes are radius. The eigenfunctions are normalized such that the max-
shown in Figs. 26d–f. It is clear that the n = 1 modes also imum horizontal velocity amplitude at the surface is 2 m s−1 ,
transport angular momentum radially upward and equatorward as inferred from observations (Löptien et al. 2018). To estimate
at higher m. However, unlike the n = 0 modes, the Reynolds FMC,θ (black dot-dashed line), we use the observational merid-
stress hvθ vθ i peaks slightly below the surface. Therefore, the cor- ional circulation data obtained by Gizon et al. (2020b). For FVD,θ
relation at the surface becomes more prominent as m increases, (black dashed line), we assume the spatially-uniform viscosity
as shown in Fig. 26f. of ν = 1012 cm2 s−1 . It is shown that the equatorward angular
It is instructive to examine how significant the angular momentum transport by the Reynolds stress is balanced by the
momentum transport by these equatorial Rossby modes can be poleward transport by meridional flow and by turbulent diffu-
in the Sun. To this end, we consider the so-called gyroscopic sion. The amplitude of FRS,θ associated with n = 1 modes are
pumping equation (e.g., Elliott et al. 2000; Miesch & Hindman found to be almost negligible, whereas that of n = 0 modes
2011) is substantial and accounts for about 30−40% of the other two
contributions FMC,θ + FVD,θ . The difference between the n = 0
∇ · (FRS + FMC + FVD ) = 0, (22) and n = 1 modes comes from that fact that the velocity eigen-
where FRS , FMC , and FVD are the angular momentum functions of the n = 1 modes peak at the surface, whereas those
fluxes transported by the Reynolds stress, meridional circu- of the n = 0 modes peak near the base. Therefore, when the
lation, and turbulent viscous diffusion, respectively. They are eigenfunctions are normalized by the surface velocity speed,
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