# Modulated rotating waves and triadic resonances in spherical fluid systems:
The case of magnetized spherical Couette flow

F. Garcia  Author to whom correspondence should be addressed: f.garcia-
<EMAIL_ADDRESS>A. Giesecke  F. Stefani

###### Abstract

The existence of triadic resonances in the magnetized spherical Couette system
(MSC) is related to the development of modulated rotating waves, which are
quasiperiodic flows understood in terms of bifurcation theory in systems with
symmetry. In contrast to previous studies in spherical geometry the resonant
modes are not inertial waves but related with the radial jet instability which
is strongly equatorially antisymmetric. We propose a general framework in
which triadic resonances are generated through successive Hopf bifurcations
from the base state. The study relies on an accurate frequency analysis of
different modes of the flow, for solutions belonging to two different
bifurcation scenarios. The azimuthal and latitudinal nonlinear coupling among
the resonant modes is analysed and interpreted using spherical harmonics and
the results are compared with previous studies in spherical geometry.

††preprint: AIP/123-QED

##  I Introduction

The onset of instabilites in rotating fluids is often related to the emergence
of a triadic resonance, which describes a resonant interaction between three
internal modes or waves that requires the fulfillment of certain selection
rules in order to take place. Such interactions occur on a small scales in the
laboratory as well as on large scales in geo- or astrophysically relevant
fluid flows. In the case of ocean flows, for example, the relevance of weak
wave resonant interactions for the global wave evolution has been investigated
for several decades (e. g. Hammack and Henderson (1993)). More recent examples
comprise the study of internal tides models of Sutherland and Jefferson
(2020), the experiments on gravity surface turbulence of Campagne  _et al._
(2019), or the theoretical study of Li  _et al._ (2020) devoted to the
analysis of steady state resonant interfacial waves in a two layer fluid
filling a duct.

Regarding celestial bodies, rotating (and electrically conducting) fluids are
responsible for the generation of magnetic fields (Jones (2011)). These flows
are assumed to be turbulent which raises the question of the initial
instabilities and the related weakly nonlinear interactions (e. g. Le Bars,
Cébron, and Le Gal (2015)). It is well known that resonant interactions are
responsible for energy transfers among different modes (e. g. Waleffe (1992);
Smith and Waleffe (1999); Galtier (2003)). Recent experiments of Brunet,
Gallet, and Cortet (2020) on wave-driven rotating turbulence have shown the
existence of a secondary instability involving four resonant waves that feed
an unforced geostrophic flow.

The triadic instabilites found in simulations and experiments of precessing
cylinders (Lagrange, Nadal, and Eloy (2011); Herault  _et al._ (2015);
Giesecke  _et al._ (2015)) have been linked to Hopf bifurcations from the
steady base state that goes along with the transition from regular (periodic
and quasiperiodic) flows to chaotic flows (Albrecht  _et al._ (2015); Lopez
and Marques (2018)).

Recently, triadic resonances were also identified in spherical systems (e. g.
Hoff, Harlander, and Triana (2016); Barik  _et al._ (2018); Lin (2021)). In
the case of mechanically driven Couette flow in a wide-gap spherical shell,
Barik  _et al._ (2018) discussed the appearance of a triadic resonance regime
in which free inertial modes form a triad with the basic non-axisymmetric
instability. For convection driven fluid flows in full spheres a triadic
resonance regime involving inertial waves was found in Lin (2021) linking
convection and mechanical drivings in agreement with the theoretical analysis
of Zhang and Liao (2017) performed in terms of inertial waves and for either
convective or mechanically forced rotating fluids systems.

Inertial waves are solutions of the inviscid Navier-Stokes equations for which
the Coriolis effects are dominant (e. g. Greenspan (1968); Rieutord and
Valdettaro (1997); Zhang  _et al._ (2001)). In the context of the spherical
Couette flow inertial wave-like motions were identified in the experimental
study of Kelley  _et al._ (2007), which provided evidence of the generation of
these type of motions when the flow is driven by strong differential rotation.
For the same problem, the study of Barik  _et al._ (2018) showed that at
higher rotation rates the fundamental instability is a shear-layer instability
(Stewartson (1966); Hollerbach  _et al._ (2004)), equatorially symmetric and
nearly geostrophic, which forms a triad with two equatorially antisymmetric
inertial modes. This instability emerges when Coriolis forces prevail (as
required for inertial geostrophic columns, see Wicht (2014)).

We demonstrate that the weakly nonlinear solutions of the magnetized spherical
Couette system (MSC) satisfy triadic resonance interactions that involve
different azimuthal and latitudinal modes. These modes are not inertial waves
but related to the radial jet instability (Hollerbach, Junk, and Egbers
(2006); Garcia and Stefani (2018); Garcia  _et al._ (2019, 2020b)), which is
basically hydrodynamic because the effects of the applied magnetic field are
weak. Contrasting to the results of Barik  _et al._ (2018) the radial jet
instability develops when viscous, rather than Coriolis, forces are dominant.
Triadic resonant flows emerge from the base state via a sequence of Hopf
bifurcations similar to the scenario described in Albrecht  _et al._ (2015)
and Lopez and Marques (2018) for the precessing cylinder. We show that this
link between triadic instability and Hopf bifurcations to periodic and
quasiperiodic flows is then valid as well in spherical geometry using the
particular example of the MSC system. The sequence of bifurcations is obtained
by modifying the magnetic field strength, though similar bifurcations are
expected when the magnitude of imposed differential rotation is increased
(Hollerbach (2009); Travnikov, Eckert, and Odenbach (2011)).

Concretely, the quasiperiodic solutions having resonant modes are modulated
rotating waves (MRW), for which a developed theory in terms of equivariant
dynamical systems exists (Rand (1982); Crawford and Knobloch (1991); Coughlin
and Marcus (1992); Golubitsky, LeBlanc, and Melbourne (2000); Golubitsky and
Stewart (2003)). These solutions have certain spatio-temporal symmetries and
appear since the system is SO$(2)\times$Z2-equivariant, i. e., invariant by
azimuthal rotations and reflections with respect to the equatorial plane. We
show that an instability in terms of a triadic resonance consisting of three
resonantly interacting modes can equivalently be described in terms of a MRW.
The spatio-temporal symmetry of the MRWs renders possible the existence of
triadic resonant relations among several modes which reflect the quadratic
nature of the nonlinear interaction.

While the mechanism proposed here is more general, it is not in contradiction
with the interpretation of triadic resonances in terms of inertial waves, as
quasiperiodic MRW involving inertial modes can be found as well in spherical
(e. g. Garcia, Chambers, and Watts (2019); Lin (2021)) and cylindrical
geometry (e. g. Albrecht  _et al._ (2018)), and their mathematical description
is analogous to that of inertial waves of Barik  _et al._ (2018) and the
references therein. We demonstrate that resonant interactions prevail on a
significant region of the parameter space and even for chaotic flows,
generated from further bifurcations of quasiperiodic MRW.

The organization of the paper is the following: In § II the model problem is
introduced and the numerical method used is briefly described. In § III we
argue how triadic resonances develop for MRWs considering the azimuthal
dependence. The identification and description of resonant triads for two
different scenarios and the comparison with previous studies is completed in §
IV while the latitudinal dependence among the resonant modes is discussed in §
V. The paper closes with § VI, summarizing the results obtained.

##  II The model equations

The magnetized spherical Couette system consist of an electrically conducting
fluid of density $\rho$, kinematic viscosity $\nu$, magnetic diffusivity
$\eta=1/(\sigma\mu_{0})$ (where $\mu_{0}$ is the magnetic permeability of the
free-space and $\sigma$ is the electrical conductivity), which fills the space
between two spheres with radius $r_{\text{i}}$ and $r_{\text{o}}$,
respectively. The gap width selected for this study is
$\chi=r_{\text{i}}/r_{\text{o}}=0.5$. The outer sphere is at rest and the
inner sphere is rotating at angular velocity $\Omega$ around the vertical axis
${\bf\hat{e}_{z}}$, and a uniform axial magnetic field of amplitude $B_{0}$ is
applied. This model has been successfully employed to compare with the very
recent experiments of Ogbonna  _et al._ (2020) in which a liquid metal within
a differentially rotating spherical vessel is subjected to an axial magnetic
field. In these experiments RWs of azimuthal symmetries $m\in\\{2,3,4\\}$ were
found in the return flow instability regime for ${\rm Re}=10^{3}$ and ${\rm
Ha}\in[27.5,40]$.

The system is described by the Navier-Stokes and induction equations scaling
the length, time, velocity and magnetic field with
$d=r_{\text{o}}-r_{\text{i}}$, $d^{2}/\nu$, $r_{\text{i}}\Omega$ and $B_{0}$,
respectively. The time dependent equations are:

$\displaystyle\partial_{t}{\mathbf{v}}+{\rm
Re}\left({\mathbf{v}}\cdot\nabla\right){\mathbf{v}}=-\nabla
p+\nabla^{2}{\mathbf{v}}+{\rm Ha}^{2}(\nabla\times{\bf
b})\times{\bf\hat{e}_{z}},$ (1) $\displaystyle
0=\nabla\times({\mathbf{v}}\times{\bf\hat{e}_{z}})+\nabla^{2}{\bf
b},\quad\nabla\cdot{\mathbf{v}}=0,\quad\nabla\cdot{\bf b}=0,$ (2)

where ${\rm Re}=\Omega r_{\text{i}}d/\nu$ is the Reynolds number, ${\rm
Ha}=B_{0}d(\sigma/(\rho\nu))^{1/2}$ is the Hartmann number, ${\mathbf{v}}$ the
velocity field and ${\bf b}$ the deviation of the magnetic field from the
axial applied field.

The inductionless approximation is used since it is valid in the limit of
small magnetic Reynolds number, ${\rm Rm}=\Omega r_{\text{i}}d/\eta\ll 1$.
This is the case for the liquid metal considered, GaInSn, with magnetic
Prandtl number ${\rm Pm}=\nu/\eta\sim O(10^{-6})$ (Plevachuk  _et al._
(2014)), at moderate ${\rm Re}=10^{3}$, since ${\rm Rm}={\rm Pm}\hskip
1.42262pt{\rm Re}\sim 10^{-3}$. The boundary conditions for the velocity field
are no-slip ($v_{r}=v_{\theta}=v_{\varphi}=0$) at $r=r_{\text{o}}$ and
constant rotation ($v_{r}=v_{\theta}=0,~{}v_{\varphi}=\sin{\theta}$, $r$,
$\theta$ and $\varphi$ being the radius, colatitude and longitude,
respectively) at $r=r_{\text{i}}$. For the magnetic field insulating boundary
conditions are applied in accordance with the experimental setups (Sisan  _et
al._ (2004); Ogbonna  _et al._ (2020)).

To solve the model equations the toroidal-poloidal formulation (Chandrasekhar
(1981)), in which the velocity field is expressed in terms of toroidal,
$\Psi$, and poloidal, $\Phi$, potentials

${\mathbf{v}}=\nabla\times\left(\Psi\mathbf{r}\right)+\nabla\times\nabla\times\left(\Phi\mathbf{r}\right),$
(3)

is used. The potentials are expressed in terms of the spherical harmonics
angular functions (truncated at order and degree $m=l=L_{\text{max}}$) and in
the radial direction a collocation method on a Gauss–Lobatto mesh of $N_{r}$
points is employed. The toroidal ($\Psi$) and poloidal ($\Phi$) potentials are
then

$\displaystyle\Psi(t,r,\theta,\varphi)=\sum_{l=0}^{L_{\text{max}}}\sum_{m=-l}^{l}{\Psi_{l}^{m}(r,t)Y_{l}^{m}(\theta,\varphi)},$
(4)
$\displaystyle\Phi(t,r,\theta,\varphi)=\sum_{l=0}^{L_{\text{max}}}\sum_{m=-l}^{l}{\Phi_{l}^{m}(r,t)Y_{l}^{m}(\theta,\varphi)},$
(5)

with $\Psi_{l}^{-m}=\overline{\Psi_{l}^{m}}$,
$\Phi_{l}^{-m}=\overline{\Phi_{l}^{m}}$, $\Psi_{0}^{0}=\Phi_{0}^{0}=0$ to
uniquely determine the two potentials, and
$Y_{l}^{m}(\theta,\varphi)=P_{l}^{m}(\cos\theta)e^{im\varphi}$, $P_{l}^{m}$
being the normalized associated Legendre functions of degree $l$ and order
$m$. The code is parallelized in the spectral and in the physical space by
using OpenMP directives. We use optimized libraries (FFTW3 Frigo and Johnson
(2005)) for the FFTs in $\varphi$ and matrix-matrix products (dgemm GOTO Goto
and van de Geijn (2008)) for the Legendre transforms in $\theta$ when
computing the nonlinear terms (see Garcia  _et al._ (2010) for further
details).

The time integration is based on high order implicit-explicit backward
differentiation formulas (IMEX–BDF) fully described in Garcia  _et al._
(2010). The nonlinear terms are treated explicitly to avoid the implicit
solution of nonlinear systems. To simplify the resolution of linear systems,
the Lorenz term is also treated explicitly. This may lead to reduced time
steps in comparison with an implicit treatment if large ${\rm Ha}$ numbers are
considered, which however is not the case of the present study with ${\rm
Ha}\sim O(1)$.

##  III Triadic resonances

The conditions for triadic resonances among inertial modes were established in
Barik  _et al._ (2018) (see their sections 2 and 7, and appendix A). Expanding
the velocity and pressure fields in terms of the Rossby number the zeroth
order of the Navier-Stokes equations leads to the inertial mode equation, with
the resonant modes being of the form
$u_{i}(t,r,\theta,\varphi)=c_{i}(t,r,\theta)\text{e}^{i(m_{i}\varphi-\omega_{i}t)}$,
$i=1,2$. The subsequent analysis of the first order equation provides the
resonance conditions, which are obtained by computing the quadratic advection
term of the sum $u_{1}+u_{2}$ (assumed to be an approximation of the zeroth-
order solution).

The time dependence of a rotating wave (RW) is described by a solid body
rotation of a fixed spatial pattern with a certain rotation frequency
$\omega$. Because of this, they are steady flows if the change of variables
$\overline{\varphi}=\varphi+\omega t$ is applied to the Navier-Stokes
equations (Eqs. (1)-(2)), see Rand (1982); Sánchez, Garcia, and Net (2013);
Garcia, Net, and Sánchez (2016); Garcia  _et al._ (2019) for a theoretical
description and applications of RWs in spherical systems. If the azimuthal
symmetry of the RW is $m_{0}$ 111If the flow has $m_{0}$-fold azimuthal
symmetry the toroidal and poloidal amplitudes of Eqs.(4,5) are nonzero only
for $m$’s which are multiples of $m_{0}$. , the spherical harmonic amplitudes
of the toroidal and poloidal potentials (Eqs. (4)-(5)) become

$\displaystyle\Psi_{l}^{m_{0}k}(r,t)=\overline{\Psi}_{l}^{m_{0}k}(r)~{}e^{-im_{0}k\omega
t},$
$\displaystyle\Phi_{l}^{m_{0}k}(r,t)=\overline{\Phi}_{l}^{m_{0}k}(r)~{}e^{-im_{0}k\omega
t},$

with $k\in\mathbb{Z}$. This is the same expression as given in Eq. (A5a,b) of
Barik  _et al._ (2018) for the inertial modes, restricted to $m=km_{0}$. The
relation between their values of the frequencies ($\omega_{j}$) of each
inertial mode and the rotation frequency of the RW waves is
$\omega_{j}=j\omega$. The above expressions for the spherical harmonic
amplitudes of the potentials evidence that RWs fulfil the trivial triadic
resonance conditions

$(i\pm j)m_{0}=im_{0}\pm jm_{0},\quad\omega_{i\pm
j}=\omega_{i}\pm\omega_{j}=(i\pm j)\omega.$ (6)

Usually RW give rise to MRW by means of Hopf-type bifurcations (see for
instance Rand (1982); Sánchez, Garcia, and Net (2013); Garcia, Net, and
Sánchez (2016); Garcia  _et al._ (2019)). We now argue how triadic resonances
can be described in terms of RWs and MRWs. Close to the bifurcation point a
MRW can be approximated by $u_{0}+\epsilon\delta u_{0}$ where $u_{0}$ is a RW
and $\delta u_{0}$ the corresponding perturbation (Floquet eigenmode, since RW
are periodic orbits, see for instance Kuznetsov (1998); Sánchez and Net
(2016); Garcia and Stefani (2018)). For the sake of simplicity only the
azimuthal dependence is considered in the following and the investigation
about the role of the latitude is reserved to § V. Consider a RW with
azimuthal symmetry $m_{0}$ and rotation frequency $\omega$ having the energy
mainly contained in the $m_{0}$ mode so it can be approximated by
$u_{o}(t,\varphi)=u_{m_{o}}\text{e}^{im_{0}(\varphi-\omega t)}$. We
approximate now the perturbation as $\delta
u_{0}(t,\varphi)=\overline{u}_{m_{1}}(t)\text{e}^{im_{1}\varphi}=u_{m_{1}}(t)\text{e}^{i(m_{1}\varphi-\omega_{1}t)}$
where $\omega_{1}$ is the new frequency provided by the Hopf bifurcation and
$m_{1}$ is the azimuthal symmetry of the Floquet mode (see Garcia  _et al._
(2019) for details). The quadratic terms $u_{o}\delta u_{0}$ appearing in the
advection term of the Navier-Stokes equations will then exite a resonant mode
since

$\text{e}^{im_{0}(\varphi-\omega
t)}\cdot\text{e}^{i(m_{1}\varphi-\omega_{1}t)}=\text{e}^{i((m_{0}+m_{1})\varphi-(m_{0}\omega+\omega_{1})t)}.$
(7)

The triadic resonance conditions will be then

$\displaystyle\omega_{2}$ $\displaystyle=$
$\displaystyle\omega_{0}+\omega_{1},\quad\text{with}\quad\omega_{0}=m_{0}\omega$
$\displaystyle m_{2}$ $\displaystyle=$ $\displaystyle m_{0}+m_{1}.$

We note that as MRW are quasiperiodic, the frequencies $\omega_{0}$ and
$\omega_{1}$ are generically incommensurable so MRW can be mathematically
expressed like solutions of the inertial equation as in Eq. (A5a,b) of Barik
_et al._ (2018). The only condition is that the axisymmetry of the base state
needs to be broken by means of a Hopf bifurcation (Ecke, Zhong, and Knobloch
(1992); Crawford and Knobloch (1991)). We also remark that if $m_{1}=m_{0}$
the nonlinear interactions of Eq. (7) are added to the self nonlinear
interactions of $u_{0}$, so the trivial resonance conditions given in Eq. (6)
may prevail.

##  IV Results

In this section we investigate triadic resonances involving different
spherical harmonic degree $m$ and order $l$ for a set of several RW and MRW
belonging to two different scenarios already studied in Garcia  _et al._
(2019, 2020b). The parameters selected for the study are an aspect ratio
$\chi=0.5$, a Reynolds number ${\rm Re}=10^{3}$, and several Hartmann numbers
with ${\rm Ha}<12.2$. For these parameters the first instability for the
steady axisymmetric base flow is nonaxisymmetric with $m=3$ azimuthal
symmetry, and antisymmetric with respect to the equator (Travnikov, Eckert,
and Odenbach (2011)). This instability is usually called “radial jet” since a
strong jet flowing outwards from the inner sphere at equatorial latitudes is
developed (Hollerbach, Junk, and Egbers (2006)).

The first bifurcated nonlinear RWs, either stable or unstable with azimuthal
symmetries $m\in{2,3,4}$, were computed by Garcia and Stefani (2018) where the
dependence of the associated rotation frequencies $\omega$ on $Ha$ on was
described. In the following we investigate resonances among different modes
that constitute a MRW. We consider several MRWs that bifurcate from the
different branches of RWs with azimuthal symmetry $m=3$ and $m=4$ (see Garcia
and Stefani (2018), Garcia  _et al._ (2019), and Garcia  _et al._ (2020b) for
further details of these MRWs).

Two different bifurcation scenarios with increasing complexity and different
sequences of azimuthal symmetry breakings are selected to strengthen the link
between different types MRW and triadic resonances. Because MRW are
quasiperiodic flows (invariant tori) they are classified according to the
number (two, three or four) of incommensurable frequencies as 2T, 3T or 4T
(“T” standing for Tori). The first bifurcation scenario provides the usual
description of stable flows (RWs and MRWs) emerging from an axisymmetric base
state. In this case the sequence of azimuthal symmetries of the flow is: $m=0$
(base state), $m=3$ (RWs), and $m=1$ (2T and 3T MRWs). For the second
scenario, a larger number of bifurcations are analyzed. In this case the
sequence of azimuthal symmetries is: $m=0$ (base state), $m=4$ (unstable RWs
and 2T MRW), $m=2$ (3T MRWs), and $m=1$ (4T MRWs and chaotic flows).
Considering this second scenario allows us to demonstrate the occurrence of
triadic resonances even for chaotic flows.

Each solution has been obtained using direct numerical simulations of Eqs.
(1-2), from which the initial transients are discarded. The spherical
harmonics series of the fields (Eqs. (4) and (5)) have been truncated at an
order and degree of $L_{\text{max}}=84$, and $N_{r}=40$ inner collocation
points are used in the radial direction. The computations have been already
validated in Garcia and Stefani (2018); Garcia  _et al._ (2020b) by increasing
$L_{\text{max}}$ and $N_{r}$ up to $126$ and $60$, respectively. To detect
resonant modes we analyse time series obtained from the poloidal potential
$\Phi$ since it is directly related with the radial velocity component

$\displaystyle v_{r}$
$\displaystyle=-\frac{1}{r\sin\theta}\left(\sin\theta\partial^{2}_{\theta\theta}\Phi+\cos\theta\partial_{\theta}\Phi+\frac{1}{\sin\theta}\partial^{2}_{\varphi\varphi}\Phi\right).$

The radial velocity component naturally reflects the structure of the
instabilities as it measures the jet outwards velocity at equatorial latitudes
and it is related with the meridional circulation developed at larger
latitudes.

###  IV.1 Identification of triadic resonances

Figure 1: (a) The time series of $\Re{\Phi^{3}_{3}(r_{d})}$ is modulated. (b)
The time series for $\Re{\Phi^{3}_{4}(r_{d})}$ is periodic. (c) The first four
frequencies of largest amplitude from the frequency spectrum of
$\Re{\Phi^{3}_{3}(r_{d})}$ (squares) and of $\Re{\Phi^{3}_{4}(r_{d})}$
(circles). The peaks selected for the resonance relations shown in table 1 are
labelled on the plot. The time series in (a) and (b) is obtained for the 2T
MRW solution at ${\rm Ha}=3.5$ which has $m=1$ azimuthal symmetry.

For analyzing the frequency spectrum it is sufficient to consider either the
poloidal or the toroidal components, since both components reflect the same
frequencies in case of regular flows (see e.g. Fig. 4a in Barik  _et al._
(2018)). Because of the symmetry, the time series of any equatorially
symmetric (resp. antisymmetric) poloidal mode, at a given radius, will exhibit
the same main peak in their frequency spectrum. That means that for each $m$
we incorporate the poloidal mode with $l=m$ (i. e. $\Phi_{m}^{m}$ which
corresponds to an equatorially symmetric term) and the poloidal mode with
$l=m+1$ (i.e. $\Phi_{m+1}^{m}$ which corresponds to an equatorially
antisymmetric term) 222We recall that a poloidal term is equatorially
symmetric if m + l is even and equatorially antisymmetric if m + l is odd.. In
order to identify triadic resonances, we specifically consider the time series
of the real part of the poloidal amplitude $R\Phi_{l}^{m}(r_{d})$, where
$r_{d}=(r_{i}+r_{o})/2$ is the middle shell radius, for several azimuthal wave
numbers $m\in{1,...,12}$ and spherical harmonic degrees $l\in{1,...,13}$.

We compute the frequency spectrum of each time series by means of Laskar’s
algorithm (Laskar (1990); Laskar, Froeschlé, and Celletti (1992); Laskar
(1993)), implemented in the SDDSToolkit (Borland  _et al._ (2017)), which is
an accurate method for the determination of the fundamental frequencies of a
time series up to prescribed tolerance. We select the frequency among the four
peaks having largest amplitude in the way we now describe. The first peak is
selected when its amplitude is significantly larger than the others. When
there are several peaks of similar magnitude, we select the frequency which
provides the larger number of resonance conditions, which is usually the
largest peak. Figure 1 exemplifies this procedure by displaying two examples
of time series and the corresponding frequency analysis for the modes
$(m,l)\in\\{(3,3),(3,4)\\}$. The time series originates from the 2T MRW with
azimuthal symmetry $m=1$ at ${\rm Ha}=3.5$ first studied in Garcia  _et al._
(2019). The time series of $R\Phi_{3}^{3}(r_{d})$ (panel (a)) is clearly
quasiperiodic with the corresponding frequency spectrum displaying two peaks
of similar magnitude (panel (c), squares) whereas the time series of
$R\Phi_{4}^{3}(r_{d})$ (panel (b)) is periodic with one single strong peak in
the spectrum (panel (c), circles).

When the frequency $f^{m}_{l}$ corresponding to each mode $(m,l)$ is
available, we seek for triadic resonances defined as

$f^{m_{0}}_{l_{0}}=f^{m_{1}}_{l_{1}}+f^{m_{2}}_{l_{2}},\quad
m_{0}=m_{1}+m_{2},$ (8)

for $m_{i}\in\\{1,...,11\\}$. The relative errors

$\epsilon_{f}=(f^{m_{0}}_{l_{0}}-f^{m_{1}}_{l_{1}}-f^{m_{2}}_{l_{2}})/f^{m_{0}}_{l_{0}}$

are computed for all the combinations satisfying $m_{0}=m_{1}+m_{2}$, and
$l_{i}=m_{i}$ or $l_{i}=m_{i}+1$ ($i=0,1,2$), with $m_{0}=3,....,12$. We
assume a resonance condition when the relative error is sufficiently small,
namely, $\epsilon_{f}<10^{-4}$. This value is similar to the accuracy obtained
for the determination of the fundamental frequency suggested by the numerical
experiments of Garcia  _et al._ (2021) performed in the same parameter region
as in the present study.

We note that there are resonances for which $l_{0}=l_{1}+l_{2}$ holds as well,
so the spherical harmonic degrees $l_{i}$ fulfill a similar condition as the
azimuthal wave numbers. In these cases the latitudinal structure imposes an
additional constraint for the emergence of a triadic resonance. Note that for
fixed $(m_{i},l_{i})$, $i=0,1,2$, obeying the condition (8), the condition
$f^{m_{0}}_{l_{0}}=f^{m_{1}}_{l_{1}+2k}+f^{m_{2}}_{l_{2}+2k}$, $k=1,2,..$
holds as well since for modes $(m_{i},l_{i}+2k)$ and $(m_{i},l_{i})$ have the
same frequency, as they have the same azimuthal wave number and equatorial
symmetry. But even if the condition $l_{0}=l_{1}+l_{2}$ is not strictly
fulfilled, the resonance relations given in Eq. (8) have to respect the
equatorial symmetry, in the sense that $l_{0}$ and $l_{1}+l_{2}$ must have the
same parity.

To check that the triad interactions preserve the same phase relation during
the whole time series a statistical measure, the bicoherence, was estimated in
Barik  _et al._ (2018); Hoff, Harlander, and Triana (2016). In our study this
is checked by directly computing the phase $\xi^{m}_{l}$ associated to the
frequencies $f^{m}_{l}$ of the interacting modes $(m,l)$, which can be easily
done with Laskar’s algorithm. The phase is computed on a specific time window
of length $T<T_{f}$, $T_{f}$ being the total length of the time series, which
is varied to obtain a time dependent value of the phase, and also of the
corresponding frequency (e. g. Garcia  _et al._ (2021)). For each resonance
relation like Eq. (8) the corresponding time dependent phases $\xi^{m}_{l}(t)$
must satisfy

$\xi^{m_{0}}_{l_{0}}(t)=\xi^{m_{1}}_{l_{1}}(t)+\xi^{m_{2}}_{l_{2}}(t),\quad
m_{0}=m_{1}+m_{2},$ (9)

up to a given tolerance.

###  IV.2 Path to triadic resonances from the base state

Figure 2: Snapshot of the poloidal amplitudes $|\Re{\Phi^{m}_{l}(r_{d})}|$,
where $r_{d}=(r_{i}+r_{o})/2$, versus the degree $l$ for $m\in\\{1,..,6\\}$.
The solution is a modulated rotating wave with $m=1$ azimuthal symmetry at
${\rm Ha}=3.5$.

This subsection details the investigation of the development of triadic
resonances among spherical harmonic modes of nonlinear solutions bifurcating
directly from the steady axisymmetric base state. At $\chi=0.5$ and ${\rm
Re}=10^{3}$ the onset of a nonaxisymmetric radial jet instability is at ${\rm
Ha}_{c}=12.2$, a value determined in Travnikov, Eckert, and Odenbach (2011)
from a linear stability analysis. The linear instability takes the form of an
equatorially antisymmetric radial jet with azimuthal symmetry $m=3$ and thus
equatorially asymmetric nonlinear rotating waves with azimuthal symmetry $m=3$
are the first time dependent flows (e. g. Ecke, Zhong, and Knobloch (1992);
Garcia and Stefani (2018)). The spherical harmonic amplitudes for the poloidal
field (Eq. (5)) are given by

$\displaystyle\Phi_{l}^{3k}(r,t)=\overline{\Phi}_{l}^{3k}(r)~{}e^{-i3k\omega
t},$ (10)

with $\omega$ being the rotation frequency of the wave and
$\overline{\Phi}_{l}^{3k}(r)$ a function which only depends on the radial
coordinate (see § III). We note that these RW have in addition a more
particular structure since $\overline{\Phi}_{l}^{3k}(r)=0$ when $l$ is odd.
This means that the mode $m=3k$ is equatorially symmetric (resp.
antisymmetric) if $k$ is even (resp. odd). Because $f^{m}_{l}=m\omega/2\pi$
these RWs satisfy the trivial triadic resonance conditions

$f^{3}_{4}=f^{9}_{10}-f^{6}_{6}=f^{12}_{12}-f^{9}_{10}=f^{15}_{16}-f^{12}_{12}=...,$

where the frequencies $f^{m}_{l}$ correspond to those of largest PSD of the
time series of $\Re{\Phi^{m}_{l}(r_{d})}$. We note that for each azimuthal
mode $m=3k$, $k=1,2,...$, there is a latitudinal mode $l$ which verifies the
resonance condition as well. For instance the relation
$f^{3}_{4}=f^{12}_{12}-f^{9}_{10}$ of above is equivalent to
$f^{3}_{4}=f^{12}_{14}-f^{9}_{10}$ since $f^{12}_{14}=f^{12}_{12}$.

Table 1: Relations between the main frequencies $f^{m}_{l}$ of the different modes $(m,l)$ for the 2T MRW with $m=1$ azimuthal symmetry at ${\rm Ha}=3.5$ and the 3T MRW with $m=1$ azimuthal symmetry at ${\rm Ha}=3.10$. These relations are satisfied up to $(f^{m_{0}}_{l_{0}}-f^{m_{1}}_{l_{1}}-f^{m_{2}}_{l_{2}})/f^{m_{0}}_{l_{0}}<\epsilon_{f}$ with $\epsilon_{f}=5\times 10^{-5}$.  $m=3$ | $m=4$ | $m=5$ | $m=6$ | $m=7$ | $m=8$ | $m=9$ | $m=10$ | $m=11$
---|---|---|---|---|---|---|---|---
$f^{3}_{3}=f^{1}_{1}+f^{2}_{2}$ | $f^{4}_{4}=f^{1}_{2}+f^{3}_{4}$ | $f^{5}_{5}=f^{1}_{1}+f^{4}_{4}$ | $f^{6}_{6}=f^{1}_{1}+f^{5}_{5}$ | $f^{7}_{7}=f^{1}_{1}+f^{6}_{6}$ | $f^{8}_{8}=f^{1}_{1}+f^{7}_{7}$ | $f^{9}_{9}=f^{1}_{1}+f^{8}_{8}$ | $f^{10}_{10}=f^{1}_{2}+f^{9}_{10}$ | $f^{11}_{11}=f^{1}_{1}+f^{10}_{10}$
| $\hskip 11.38109pt=2f^{2}_{3}$ | $\hskip 11.38109pt=f^{1}_{2}+f^{4}_{5}$ | $\hskip 11.38109pt=f^{1}_{2}+f^{5}_{6}$ | $\hskip 11.38109pt=f^{2}_{2}+f^{5}_{5}$ | $\hskip 11.38109pt=f^{2}_{2}+f^{6}_{6}$ | $\hskip 11.38109pt=f^{2}_{2}+f^{7}_{7}$ | $\hskip 14.22636pt=f^{2}_{3}+f^{8}_{9}$ | $\hskip 14.22636pt=f^{1}_{2}+f^{10}_{11}$
|  | $\hskip 11.38109pt=f^{2}_{3}+f^{3}_{4}$ | $\hskip 11.38109pt=f^{2}_{2}+f^{4}_{4}$ | $\hskip 11.38109pt=f^{2}_{3}+f^{5}_{6}$ | $\hskip 11.38109pt=f^{2}_{3}+f^{6}_{7}$ | $\hskip 11.38109pt=f^{3}_{3}+f^{6}_{6}$ | $\hskip 14.22636pt=f^{3}_{4}+f^{7}_{8}$ | $\hskip 14.22636pt=f^{2}_{3}+f^{9}_{10}$
|  |  | $\hskip 11.38109pt=f^{2}_{3}+f^{4}_{5}$ | $\hskip 11.38109pt=f^{3}_{3}+f^{4}_{4}$ | $\hskip 11.38109pt=f^{3}_{3}+f^{5}_{5}$ | $\hskip 11.38109pt=f^{4}_{5}+f^{5}_{6}$ | $\hskip 14.22636pt=f^{4}_{4}+f^{6}_{6}$ | $\hskip 14.22636pt=f^{3}_{4}+f^{8}_{9}$
|  |  | $\hskip 11.38109pt=2f^{3}_{4}$ | $\hskip 11.38109pt=f^{3}_{4}+f^{4}_{5}$ | $\hskip 11.38109pt=f^{3}_{4}+f^{5}_{6}$ |  | $\hskip 14.22636pt=f^{4}_{5}+f^{6}_{7}$ | $\hskip 14.22636pt=f^{4}_{4}+f^{7}_{7}$
|  |  |  |  | $\hskip 11.38109pt=2f^{4}_{5}$ |  | $\hskip 14.22636pt=2f^{5}_{5}$ | $\hskip 14.22636pt=f^{4}_{5}+f^{7}_{8}$
|  |  |  |  |  |  |  | $\hskip 14.22636pt=f^{5}_{5}+f^{6}_{6}$
|  |  |  |  |  |  |  | $\hskip 14.22636pt=f^{5}_{6}+f^{6}_{7}$
$f^{3}_{4}=f^{1}_{1}+f^{2}_{3}$ | $f^{4}_{5}=f^{1}_{1}+f^{3}_{4}$ | $f^{5}_{6}=f^{1}_{1}+f^{4}_{5}$ | $f^{6}_{7}=f^{1}_{2}+f^{5}_{5}$ | $f^{7}_{8}=f^{1}_{1}+f^{6}_{7}$ | $f^{8}_{9}=f^{1}_{1}+f^{7}_{8}$ | $f^{9}_{10}=f^{1}_{2}+f^{8}_{8}$ | $f^{10}_{11}=f^{1}_{1}+f^{9}_{10}$ | $f^{11}_{12}=f^{1}_{1}+f^{10}_{11}$
$\hskip 11.38109pt=f^{1}_{2}+f^{2}_{2}$ | $\hskip 11.38109pt=f^{1}_{2}+f^{3}_{3}$ | $\hskip 11.38109pt=f^{2}_{3}+f^{3}_{3}$ | $\hskip 11.38109pt=f^{2}_{3}+f^{4}_{4}$ | $\hskip 11.38109pt=f^{1}_{2}+f^{6}_{6}$ | $\hskip 14.22636pt=f^{1}_{2}+f^{7}_{7}$ | $\hskip 14.22636pt=f^{1}_{1}+f^{8}_{9}$ | $\hskip 14.22636pt=f^{1}_{2}+f^{9}_{9}$ | $\hskip 14.22636pt=f^{2}_{2}+f^{9}_{10}$
| $\hskip 11.38109pt=f^{2}_{2}+f^{2}_{3}$ | $\hskip 11.38109pt=f^{2}_{2}+f^{3}_{4}$ |  | $\hskip 11.38109pt=f^{2}_{3}+f^{5}_{5}$ | $\hskip 14.22636pt=f^{2}_{2}+f^{6}_{7}$ | $\hskip 14.22636pt=f^{2}_{3}+f^{7}_{7}$ | $\hskip 14.22636pt=f^{2}_{2}+f^{8}_{9}$ | $\hskip 14.22636pt=f^{2}_{3}+f^{9}_{9}$
|  |  |  | $\hskip 11.38109pt=f^{3}_{4}+f^{4}_{4}$ | $\hskip 14.22636pt=f^{2}_{3}+f^{6}_{6}$ | $\hskip 14.22636pt=f^{2}_{2}+f^{7}_{8}$ | $\hskip 14.22636pt=f^{2}_{3}+f^{8}_{8}$ | $\hskip 14.22636pt=f^{3}_{3}+f^{8}_{9}$
|  |  |  |  | $\hskip 14.22636pt=f^{3}_{4}+f^{5}_{5}$ | $\hskip 14.22636pt=f^{3}_{4}+f^{6}_{6}$ | $\hskip 14.22636pt=f^{3}_{3}+f^{7}_{8}$ | $\hskip 14.22636pt=f^{3}_{4}+f^{8}_{8}$
|  |  |  |  | $\hskip 14.22636pt=f^{4}_{4}+f^{4}_{5}$ | $\hskip 14.22636pt=f^{3}_{3}+f^{6}_{7}$ | $\hskip 14.22636pt=f^{3}_{4}+f^{7}_{7}$ | $\hskip 14.22636pt=f^{4}_{5}+f^{7}_{7}$
|  |  |  |  |  | $\hskip 14.22636pt=f^{4}_{5}+f^{5}_{5}$ | $\hskip 14.22636pt=f^{4}_{5}+f^{6}_{6}$ | $\hskip 14.22636pt=f^{5}_{6}+f^{6}_{6}$
|  |  |  |  |  | $\hskip 14.22636pt=f^{4}_{4}+f^{5}_{6}$ | $\hskip 14.22636pt=f^{5}_{5}+f^{5}_{6}$ |

According to Garcia and Stefani (2018) the stability region of RWs with $m=3$
azimuthal symmetry extends down to ${\rm Ha}=3.95$ and a secondary Hopf
bifurcation gives rise to quasiperiodic 2T MRWs with azimuthal symmetry $m=1$.
For these types of solutions several resonance conditions, reflecting the
nonlinear interactions among the modes, are found. To provide an idea of the
contributions of the different modes for a selected MRW at ${\rm Ha}=3.5$,
figure 2 displays the poloidal amplitudes $|\Re{\Phi^{m}_{l}(r_{d})}|$ at a
fixed time instant for different values of $m$ and $l$ (see figure caption).
In contrast to the special structure of RWs, where the equatorial symmetry of
the mode $m$ depends on its parity, this figure demonstrates that each mode
$m$ is equatorially asymmetric, as all the degree $l\geq m$ are nonzero.
However, the main structure of the RWs still prevails as the dominant mode
$m=3$ has a significantly larger equatorially antisymmetric component and for
the $m=6$ mode the largest component is the equatorially symmetric. The
$m=2,4,5$ modes have a dominant equatorially antisymmetric ($m=2,4$) or
symmetric ($m=5$) component but both components are of similar magnitude in
the case of $m=1$. The patterns of the radial velocity for the different
components of this solution were analyzed in Garcia  _et al._ (2019)
(concretely Figs. 6 and 7) where the interested reader will find a
comprehensive description.

Following the procedure described in § IV.1 we have identified a multitude of
triadic resonance relations among different modes of the 2T MRW at ${\rm
Ha}=3.5$. The relations are listed in table 1. The first row of this table
displays the resonance relations
$f^{m_{0}}_{m_{0}}=f^{m_{1}}_{l_{1}}+f^{m_{2}}_{l_{2}}$, for each
$m_{0}\in\\{3,...,11\\}$ from left to right, corresponding to the equatorially
symmetric mode $(m,l)=(m_{0},m_{0})$. The second row displays
$f^{m_{0}}_{m_{0}+1}=f^{m_{1}}_{l_{1}}+f^{m_{2}}_{l_{2}}$, which corresponds
to the resonance relation for the equatorially antisymmetric mode
$(m,l)=(m_{0},m_{0}+1)$.

We note that for all the frequencies $f^{m_{0}}_{m_{0}}$ of the equatorially
symmetric modes, with the exception of $m_{0}=4,10$, a resonance relation
satisfying $f^{m_{0}}_{m_{0}}=f^{1}_{1}+f^{m_{0}-1}_{m_{0}-1}$ can be found,
which means that both the azimuthal and latitudinal wave numbers are resonant.
For $m_{0}=4,10$ the resonance relation involving azimuthal and latitudinal
wave numbers is extracted from those of $m_{0}=5,11$, i. e.,
$f^{4}_{4}=f^{5}_{5}-f^{1}_{1}$ and $f^{10}_{10}=f^{11}_{11}-f^{1}_{1}$. The
same happens for the antisymmetric modes where either the relations
$f^{m_{0}}_{m_{0}+1}=f^{1}_{1}+f^{m_{0}-1}_{m_{0}}$ or
$f^{m_{0}}_{m_{0}+1}=f^{1}_{2}+f^{m_{0}-1}_{m_{0}-1}$ can be found.

It is interesting to remark that 2T MRWs with $m=1$ azimuthal symmetry arise
from RWs with $m=3$ symmetry by applying perturbations (Floquet modes, already
computed in Garcia and Stefani (2018)) of azimuthal symmetry $m=1$. Then, the
nonlinear coupling (described in § III) is responsible for the resonant
interactions described for the 2T MRW at ${\rm Ha}=3.5$ for which the modes
$(m,l)=\\{(1,1),(1,2)\\}$ are in resonance with all the other modes
investigated in Table 1. The triadic resonance relations found for the 2T MRW
at ${\rm Ha}=3.5$ have been found also for 2T MRWs belonging to the same
branch which is stable for ${\rm Ha}\in[3.4,3.94]$. In addition, we have also
found that the resonance relations of Table 1 are conserved for 3T MRWs that
bifurcates from the 2T MRW at ${\rm Ha}=3.395$ and which are stable down to
${\rm Ha}=3.1$. Then, the resonance interactions take place for all solutions
obtained in the range ${\rm Ha}\in[3.4,3.94]$, which is the range of stability
of periodic and quasiperiodic flows which bifurcate from the base state.

Figure 3: Analysis for a 2T MRW with $m=1$ azimuthal symmetry at ${\rm
Ha}=3.5$. (a) Leading frequencies and amplitudes of the time series of
poloidal component of the different modes $(m,l)$. The frequency $f^{m}_{l}$
corresponds to the main peak of the frequency spectrum of the time series of
$\Re{\Phi^{m}_{l}(r_{d})}$. (b) Phase coupling among the different modes
corresponding to different types of resonances shown in Table 1. The phase of
each frequency $f^{m}_{l}$ is $\xi^{m}_{l}$. (c) The coupling
$\xi^{3}_{4}=\xi^{1}_{1}+\xi^{2}_{3}$ is investigated by a time-dependent
frequency analysis with time windows of length $T$.

Figure 3(a) shows the frequencies $f^{m}_{l}$ involved in Table 1 versus their
corresponding amplitudes. The relations
$f^{m_{0}}_{m_{0}}<f^{m_{0}+1}_{m_{0}+1}$ and
$f^{m_{0}}_{m_{0}+1}<f^{m_{0}+1}_{m_{0}+2}$ are verified for all
$m_{0}=1,..,11$ meaning that low wave numbers have smaller frequencies (slow
modes) and larger wave numbers have larger frequencies (fast modes). The
figure shows however that no clear threshold between slow and fast modes can
be defined. In contrast to the results in Barik  _et al._ (2018), where only
resonances between one slow and two fast modes are reported, we find
resonances between slow modes (e. g. $f^{3}_{3}=f^{1}_{1}+f^{2}_{2}$), fast
and slow modes (e. g. $f^{11}_{11}=f^{1}_{1}+f^{10}_{10}$), and fast modes (e.
g. $f^{10}_{10}=f^{4}_{4}+f^{6}_{6}$).

Figures 3(b) and (c) correspond to the analysis of the phase coupling among
the resonant modes, described in § IV.1. Concretely, the time dependent
relations given in Eq. (9) are displayed in Fig. 3(b) for some resonant
triads. The time window used for this figure is $T=2$ over a time series
covering the interval $[0,15]$. The phase coupling is satisfied up to an
accuracy which is similar to that for the frequency determination as it is
demonstrated in Fig. 3(c). The latter figure shows that the tiny variation of
$\xi^{3}_{4}-\xi^{1}_{1}-\xi^{2}_{3}$ decreases as the size of the time window
$T$ increases, i. e. as the accuracy of the frequency determination increases.

###  IV.3 Path to triadic resonances from unstable rotating waves

Unstable RWs with azimuthal symmetry $m=4$ bifurcate from the axisymmetric
base state at a Hartmann number ${\rm Ha}=11.04<{\rm Ha}_{c}=12.2$ (see Garcia
and Stefani (2018)), where ${\rm Ha}_{c}$ is the critical value predicted by
the linear theory (Travnikov, Eckert, and Odenbach (2011)). These RWs again
correspond to the radial jet instability. Because the linear eigenfunctions
are equatorially antisymmetric, and RWs are nonlinear solutions, these RWs are
equatorially asymmetric. They have however a particular structure (as happened
for RWs described in § IV.2) since the $m=4k$ modes are equatorially
antisymmetric (resp. symmetric) when $k$ is an odd (resp. even) integer
number. Then, if $\omega$ is the rotation frequency of the RW the spherical
harmonic amplitudes for the poloidal field of Eq. (5) read

$\displaystyle\Phi_{l}^{4k}(r,t)=\overline{\Phi}_{l}^{4k}(r)~{}e^{-i4k\omega
t},$ (11)

with $\overline{\Phi}_{l}^{4k}(r)=0$ when $k+l$ is an odd integer number.

A secondary Hopf bifurcation (see Garcia  _et al._ (2020b) gives rise to 2T
MRWs with the same azimuthal symmetry and the same mode structure. In this
case the spherical harmonic amplitudes $\Phi_{l}^{4k}(r,t)$ become
quasiperiodic of the form

$\displaystyle\Phi_{l}^{4k}(r,t)=\overline{\Phi}_{l}^{4k}(t,r)~{}e^{-i4k\omega
t},$ (12)

with $\overline{\Phi}_{l}^{4k}(t,r)$ being a periodic function with
$\overline{\Phi}_{l}^{4k}(t,r)=0$ when $k+l$ is an odd integer number, and
$\omega$ the rotation frequency of the MRW (e. g. Rand (1982)).

Because of these structures, both RWs and MRWs, fulfill the triadic resonance
conditions for the azimuthal wave number $m$:

$f^{4}_{5}=f^{12}_{13}-f^{8}_{8}=f^{16}_{16}-f^{12}_{13}=f^{20}_{21}-f^{16}_{16}=...,$
(13)

where the frequencies $f^{m}_{l}$ correspond to those of largest amplitude in
the frequency spectrum of the time series of $\Re{\Phi^{m}_{l}(r_{d})}$. As in
the previous section, for each azimuthal mode $m=4k$, $k=1,2,...$, there is a
latitudinal mode $l$ which verify the resonance condition. For instance the
relation $f^{4}_{5}=f^{16}_{16}-f^{12}_{13}$ given above corresponds to
$f^{4}_{5}=f^{16}_{18}-f^{12}_{13}$, since $f^{16}_{18}=f^{16}_{16}$.

By further varying the Hartmann number a tertiary Hopf bifurcation gives rise
to 3T MRWs, i. e. quasiperiodic flows with three incommensurable frequencies.
By means of this bifurcation the $m=4$ azimuthal symmetry is broken and the
new MRWs acquire azimuthal symmetry $m=2$. In contrast to MRWs with $m=4$
azimuthal symmetry, all the azimuthal modes $m$ are now equatorially
asymmetric, meaning that
$\Phi^{m}_{m}(t,r),\Phi^{m}_{m+1}(t,r),....,\Phi^{m}_{L_{\text{max}}}(t,r)$
are nonzero. The dominant azimuthal wave numbers for this solution are
$m=4,2,8,6,10,12$ (see Fig. (5) of Garcia  _et al._ (2020b)) so we focus on
studying the resonances among these modes. We have identified several
relations between the frequencies $f^{m}_{l}$, which are listed in Table 2.
These relations include those of Eq. (13) found for the parent RWs and MRWs.
The fact that bifurcated solutions conserve the resonance relations was also
found in the scenario studied in § IV.2. As will be evidenced in the following
this is indeed true for further bifurcations found in Garcia  _et al._
(2020b).

Table 2: Relations between the main frequencies $f^{m}_{l}$ of the different modes $(m,l)$ for the 3T MRW with $m=2$ azimuthal symmetry at ${\rm Ha}=1.6$. $m=4$ | $m=6$ | $m=8$ | $m=10$ | $m=12$
---|---|---|---|---
$f^{4}_{4}=2f^{2}_{3}$ | $f^{6}_{6}=f^{2}_{2}+f^{4}_{4}=f^{2}_{3}+f^{4}_{5}$ | $f^{8}_{8}=f^{2}_{2}+f^{6}_{6}=f^{2}_{3}+f^{6}_{7}=2f^{4}_{5}$ | $f^{10}_{10}=f^{2}_{2}+f^{8}_{8}=f^{4}_{5}+f^{6}_{7}$ | $f^{12}_{12}=f^{2}_{2}+f^{10}_{10}=2f^{6}_{7}$
$f^{4}_{5}=f^{2}_{2}+f^{2}_{3}$ | $f^{6}_{7}=f^{2}_{2}+f^{4}_{5}$ | $f^{8}_{9}=f^{2}_{3}+f^{6}_{6}=f^{4}_{4}+f^{4}_{5}$ | $f^{10}_{11}=f^{2}_{2}+f^{8}_{9}=f^{2}_{3}+f^{8}_{8}$ | $f^{12}_{13}=f^{2}_{2}+f^{10}_{11}=f^{2}_{3}+f^{10}_{10}$
|  |  | $\hskip 14.22636pt=f^{4}_{4}+f^{6}_{7}=f^{4}_{5}+f^{6}_{6}$ | $\hskip 14.22636pt=f^{4}_{5}+f^{8}_{8}=f^{6}_{6}+f^{6}_{7}$

After a quaternary Hopf bifurcation, totally breaking the azimuthal symmetry,
quasiperiodic MRWs with four incommensurable frequencies develop and the
resonance relations for the same 4T MRW analyzed in Garcia  _et al._ (2020b),
at ${\rm Ha}=1.4$, are provided in Table 3. There are less relations when
comparing with Table 1 of § IV.2, obtained for a 2T MRW, which seems
reasonable since the time dependence is now more complex as illustrated in
figure 4(a,b). The time series of $\Re{\Phi^{m}_{l}(r_{d})}$ have now several
comparable peaks in their frequency spectrum so the selection of the frequency
$f^{m}_{l}$ becomes less clear (compare Fig. 1(c) with Fig. 4(c)).

Table 3: Relations between the main frequencies $f^{m}_{l}$ of the different modes $(m,l)$ for the 4T MRW with $m=1$ azimuthal symmetry at ${\rm Ha}=1.4$ and a chaotic flow with $m=1$ azimuthal symmetry at ${\rm Ha}=0.5$. For $m=3$ only the relation $f^{3}_{3}=f^{1}_{1}+f^{2}_{2}$ has been found. Note that these relations contain those for the 3T MRW with $m=2$ azimuthal symmetry shown in Table 2. These relations are satisfied up to $(f^{m_{0}}_{l_{0}}-f^{m_{1}}_{l_{1}}-f^{m_{2}}_{l_{2}})/f^{m_{0}}_{l_{0}}<\epsilon_{f}$ with $\epsilon_{f}=4\times 10^{-5}$ for the 4T MRW and with $\epsilon_{f}=1.5\times 10^{-4}$ for the chaotic flow. $m=4$ | $m=5$ | $m=6$ | $m=7$ | $m=8$ | $m=9$ | $m=10$ | $m=11$ | $m=12$
---|---|---|---|---|---|---|---|---
$f^{4}_{4}=2f^{2}_{3}$ | $f^{5}_{5}=f^{2}_{3}+f^{3}_{4}$ | $f^{6}_{6}=f^{1}_{1}+f^{5}_{5}$ | $f^{7}_{7}=f^{2}_{2}+f^{5}_{5}$ | $f^{8}_{8}=f^{1}_{1}+f^{7}_{7}$ | $f^{9}_{9}=f^{2}_{2}+f^{7}_{7}$ | $f^{10}_{10}=f^{1}_{1}+f^{9}_{9}$ | $f^{11}_{11}=f^{1}_{1}+f^{10}_{10}$ | $f^{12}_{12}=f^{2}_{2}+f^{10}_{10}$
|  | $\hskip 11.38109pt=f^{2}_{2}+f^{4}_{4}$ | $\hskip 11.38109pt=f^{2}_{3}+f^{5}_{6}$ | $\hskip 11.38109pt=f^{2}_{2}+f^{6}_{6}$ | $\hskip 11.38109pt=f^{3}_{4}+f^{6}_{7}$ | $\hskip 14.22636pt=f^{2}_{2}+f^{8}_{8}$ | $\hskip 14.22636pt=f^{3}_{3}+f^{8}_{8}$ | $\hskip 14.22636pt=f^{3}_{3}+f^{9}_{9}$
|  | $\hskip 11.38109pt=f^{2}_{3}+f^{4}_{5}$ | $\hskip 11.38109pt=f^{3}_{4}+f^{4}_{5}$ | $\hskip 11.38109pt=f^{2}_{3}+f^{6}_{7}$ | $\hskip 11.38109pt=f^{4}_{5}+f^{5}_{6}$ | $\hskip 14.22636pt=f^{3}_{3}+f^{7}_{7}$ | $\hskip 14.22636pt=f^{4}_{5}+f^{7}_{8}$ | $\hskip 14.22636pt=f^{5}_{6}+f^{7}_{8}$
|  |  |  | $\hskip 11.38109pt=f^{3}_{3}+f^{5}_{5}$ |  | $\hskip 14.22636pt=f^{3}_{4}+f^{7}_{8}$ |  | $\hskip 14.22636pt=2f^{6}_{7}$
|  |  |  | $\hskip 11.38109pt=2f^{4}_{5}$ |  | $\hskip 14.22636pt=f^{4}_{5}+f^{6}_{7}$ |  |
$f^{4}_{5}=f^{1}_{1}+f^{3}_{4}$ | $f^{5}_{6}=f^{2}_{2}+f^{3}_{4}$ | $f^{6}_{7}=f^{1}_{1}+f^{5}_{6}$ | $f^{7}_{8}=f^{1}_{1}+f^{6}_{7}$ | $f^{8}_{9}=f^{2}_{3}+f^{6}_{6}$ | $f^{9}_{10}=f^{2}_{3}+f^{7}_{7}$ | $f^{10}_{11}=f^{1}_{1}+f^{9}_{10}$ | $f^{11}_{12}=f^{2}_{2}+f^{9}_{10}$ | $f^{12}_{13}=f^{1}_{1}+f^{11}_{12}$
$\hskip 11.38109pt=f^{2}_{2}+f^{2}_{3}$ |  | $\hskip 11.38109pt=f^{2}_{2}+f^{4}_{5}$ | $\hskip 11.38109pt=f^{3}_{3}+f^{4}_{5}$ | $\hskip 11.38109pt=f^{4}_{4}+f^{4}_{5}$ | $\hskip 14.22636pt=f^{3}_{4}+f^{6}_{6}$ | $\hskip 14.22636pt=f^{2}_{2}+f^{8}_{9}$ | $\hskip 14.22636pt=f^{2}_{3}+f^{9}_{9}$ | $\hskip 14.22636pt=f^{2}_{2}+f^{10}_{11}$
|  | $\hskip 11.38109pt=f^{3}_{3}+f^{3}_{4}$ |  |  | $\hskip 14.22636pt=f^{4}_{5}+f^{5}_{5}$ | $\hskip 14.22636pt=f^{2}_{3}+f^{8}_{8}$ | $\hskip 14.22636pt=f^{3}_{4}+f^{8}_{8}$ | $\hskip 14.22636pt=f^{2}_{3}+f^{10}_{10}$
|  |  |  |  | $\hskip 14.22636pt=f^{4}_{4}+f^{5}_{6}$ | $\hskip 14.22636pt=f^{4}_{4}+f^{6}_{7}$ | $\hskip 14.22636pt=f^{4}_{5}+f^{7}_{7}$ | $\hskip 14.22636pt=f^{3}_{3}+f^{9}_{10}$
|  |  |  |  |  | $\hskip 14.22636pt=f^{4}_{5}+f^{6}_{6}$ | $\hskip 14.22636pt=f^{5}_{5}+f^{6}_{7}$ | $\hskip 14.22636pt=f^{4}_{5}+f^{8}_{8}$
|  |  |  |  |  |  | $\hskip 14.22636pt=f^{5}_{6}+f^{6}_{6}$ | $\hskip 14.22636pt=f^{5}_{5}+f^{7}_{8}$
|  |  |  |  |  |  |  | $\hskip 14.22636pt=f^{6}_{6}+f^{6}_{7}$
Figure 4: (a) The time series of $\Re{\Phi^{4}_{4}(r_{d})}$ is strongly
modulated. (b) The time series for $\Re{\Phi^{4}_{5}(r_{d})}$ is slightly
modulated. (c) The first ten frequencies of largest amplitude from the
frequency spectrum of $\Re{\Phi^{4}_{4}(r_{d})}$ (squares) and of
$\Re{\Phi^{4}_{5}(r_{d})}$ (circles). The peaks selected for the resonance
relations shown in table 3 are labelled on the plot. The solution is a 4T MRW
with $m=1$ azimuthal symmetry at ${\rm Ha}=1.4$.

The resonance relations found for the 3T MRWs prevail for the bifurcated 4T
MRWs (all relations of Table 2 are included in Table 3). Moreover, the
resonance relations for the 4T MRWs of Table 3 even prevail for chaotic flows
bifurcated from these 4T MRWs. We have checked this by considering a chaotic
flow at ${\rm Ha}=0.5$. As noticed in Garcia  _et al._ (2020a, b) the main
frequencies associated to the flow and to the volume-averaged properties
remain nearly constant for a wide interval of Hartmann numbers, at which
several types of 2T, 3T and 4T MRW can be found. Figure 5 evidences that this
is also true for the frequencies $f^{m}_{l}$. Figure 5 shows from top to
bottom the selected frequencies $f^{m}_{l}$ and amplitudes for the sequence of
bifurcated solutions: 2T MRW and 3T MRW both at ${\rm Ha}=1.6$ (panel (a)), 4T
MRW at ${\rm Ha}=1.4$ (panel (b)), and a chaotic flow at ${\rm Ha}=0.5$ (panel
(c)). From this figure it is clear that $f^{m}_{l}$ are almost unchanged.

Figure 5: Leading frequencies and amplitudes of the time series of poloidal
component of the different modes $(m,l)$. The frequency $f^{m}_{l}$
corresponds to one peak of the frequency spectrum of the time series of
$\Re{\Phi^{m}_{l}(r_{d})}$, where $r_{d}=(r_{i}+r_{o})/2$. (a) The solutions
are a 2T MRW with $m=4$ azimuthal symmetry (squares) and a 3T MRW with $m=2$
azimuthal symmetry (crosses), both at ${\rm Ha}=1.6$. (b) The solution is a 4T
MRW with $m=1$ azimuthal symmetry (circles) at ${\rm Ha}=1.4$. (c) The
solution is a chaotic wave with $m=1$ azimuthal symmetry (triangles) at ${\rm
Ha}=0.5$.

Figure 6 is concerned with the analysis of the phase coupling among the
resonant modes (see § IV.1) for the 4T MRW (top row) at ${\rm Ha}=1.4$ and for
the chaotic flow at ${\rm Ha}=0.5$ (bottom row). In this bifurcation scenario
the time dependence of the phase coupling relations
$\xi^{m_{0}}_{l_{0}}-\xi^{m_{1}}_{l_{1}}-\xi^{m_{2}}_{l_{2}}$, although
noticeable, is being damped as the time window $T$ increases. This means that
for sufficiently large time windows the phase coupling tends to be verified.
For the 4T and chaotic flow, the necessity of considering very large time
windows ($T>10$) in time dependent frequency analysis has been already pointed
out in Garcia  _et al._ (2020b).

Figure 6: Phase coupling among the different modes corresponding to two
different types of resonances $f^{4}_{5}=f^{1}_{1}+f^{3}_{4}$ and
$f^{6}_{6}=f^{2}_{3}+f^{4}_{5}$ shown in Table 3. The phase of each frequency
$f^{m}_{l}$ is $\xi^{m}_{l}$. The coupling is investigated by a time-dependent
frequency analysis with time windows of length $T$. (a,c) The relation
$\xi^{4}_{5}(t)-\xi^{1}_{1}(t)-\xi^{3}_{4}(t)$ tends to be roughly constant as
the length of the time window $T$ increases. (b,d) The relation
$\xi^{6}_{6}(t)-\xi^{2}_{3}(t)-\xi^{4}_{5}(t)$ is nearly constant as the
length of the time window $T$ increases. (a,b) Correspond to a 4T MRW at ${\rm
Ha}=1.4$ and (c,d) correspond to a chaotic wave at ${\rm Ha}=0.5$.

###  IV.4 Comparison with triadic resonances of inertial modes

We now compare quantitatively our results with the previous studies of Barik
_et al._ (2018) and Lin (2021) in rotating spherical geometry for which
triadic resonances, involving inertial modes, are described. In the first
place we compare with the experiments and simulations of Barik  _et al._
(2018) as the analyzed problem, the SC flow with a rotating inner (with
$\Omega_{i}$) and outer (with $\Omega_{o}$) spheres, is a mechanically driven
system equivalent to the problem considered here, the MSC flow. In the second
place, we compare with the thermal convection problem in a full sphere of Lin
(2021). Although we are contrasting problems with different types of driving
or dynamical regimes the comparison makes sense since all the systems are
SO$(2)\times$Z2-equivariant.

The frequencies shown in Table 1 and fig. 3(a) are dimensionless. With the
time scale $d^{2}/\nu$ considered in our modelling, dimensional frequencies,
$f_{d}$, are obtained by $f_{d}=f\nu/d^{2}$ so

$\frac{\omega_{d}}{\Omega_{i}}=\frac{2\pi
f_{d}}{\Omega_{i}}=\frac{\omega}{{\rm Re}(\chi-1)},$

with $\omega=2\pi f$. The ratios $\omega_{d}/\Omega_{i}$ are the angular
frequences normalized by the angular frequency of rotation of the inner sphere
since the outer sphere is at rest, $\Omega_{o}=0$. For the modes analyzed in
fig. 3(a) the slowest mode is $(m,l)=(1,1)$ with a ratio
$\omega_{d}/\Omega_{i}=0.097$. This value is very similar to that of the basic
$m=1$ instability studied in Barik  _et al._ (2018) with a ratio
$\omega_{d}/\Omega_{o}=0.086$. In our case the fastest mode corresponds to
$(m,l)=(11,11)$ with a ratio $\omega_{d}/\Omega_{i}=3.22$. The mode
$(m,l)=(3,4)$, which has the largest peak in the frequency spectrum, has a
ratio close to unity $\omega_{d}/\Omega_{i}=0.83$. In the case of the SC flow
(Barik  _et al._ (2018)) values in the range $(0.4,1)$ have been obtained for
the ratio $\omega_{d}/\Omega_{o}$ in the case of fast inertial modes.

The development of triadic resonances for rotating thermal convection in a
full sphere studied by Lin (2021) follows the same picture as investigated
here. Rotating waves in the form of inertial modes with $m=4$ (e. g. Zhang
(1993); Net, Garcia, and Sánchez (2008); Sánchez, Garcia, and Net (2016)) are
the first time-dependent instabilities in the considered parameter regime (low
Prandtl and Ekman numbers). Lin (2021) slightly increases the forcing
parameter (the Rayleigh number) to obtain an azimuthally asymmetric
quasiperiodic flow with two frequencies (see their Fig. 1), i. e. a 2T MRW
with azimuthal symmetry $m=1$, for which resonant azimuthal modes have been
found. The appearance of MRWs in rotating thermal convection in spherical
shells due to a Hopf bifurcation from RWs is a generic scenario which has been
studied in Garcia  _et al._ (2015); Garcia, Net, and Sánchez (2016). In
addition, and in agreement with our results, Lin (2021) has also found
resonant behaviour for chaotic flows close to the onset.

For the inertial convective resonant solutions the value $\omega_{d}/\Omega$
has been also computed (see Table 1 of Lin (2021)) to identify fast and slow
modes. By varying the Ekman and Prandtl numbers it was found that
$|\omega_{d}/\Omega|\in(0.0899,0.5352$, either for the primary or free modes,
which are $m=\\{4,1,3\\}$, $m=\\{1,7,8\\}$, and $m=\\{3,10,13\\}$ for the
three different cases studied in Table 1 of Lin (2021). These results are
indeed similar to the results for the radial jet instability (Secs. IV.2 and
IV.3) since resonances involving these azimuthal wave numbers are found with
values $\omega_{d}/\Omega_{i}$ of similar magnitude.

##  V Nonlinear interactions among latitudinal modes

Figure 7: Contour plots on spherical, latitudinal and meridional sections
through a local maximum for the $(m,l)=(3,3)$ component of the nonlinear term
corresponding to the Navier-Stokes equations for the poloidal field. The
evaluation of this nonlinear term is applied to the poloidal field whereas the
toroidal field is set to zero. All the spherical harmonics amplitudes are
considered for the poloidal field in the top row but only the modes
$(m,l)=(1,1),(2,2)$ are considered in the bottom row. The solution is a 2T MRW
with $m=1$ azimuthal symmetry at ${\rm Ha}=3.5$.

Figure 8: As figure 7 but for the $(m,l)=(3,4)$ component of the nonlinear
term. In this figure the modes $(m,l)=(1,2),(2,2)$ and $(m,l)=(1,1),(2,3)$ are
considered in the middle and bottom row. The solution is a 2T MRW with $m=1$
azimuthal symmetry at ${\rm Ha}=3.5$.

The quadratic nonlinear nature of the advection term in the Navier-Stokes
equations provides a way to to understand the development of the different
resonances obtained in § IV.2 and § IV.3. The azimuthal coupling among the
different resonant waves has been already described in § III and we now focus
on the latitudinal coordinate.

The product of the spherical harmonics expansion (e. g. Eq. (5)) gives rise to
products of associated Legendre functions
$P^{m_{1}}_{l_{1}}(\cos{\theta})P^{m_{2}}_{l_{2}}(\cos{\theta})\equiv
P^{m_{1}}_{l_{1}}P^{m_{2}}_{l_{2}}$ which can be expressed in terms of a
linear combination of associated Legendre functions of order $m_{1}+m_{2}$ and
appropriate degree $l$ (see Hwang (1995)):

$P^{m_{1}}_{l_{1}}P^{m_{2}}_{l_{2}}=\sum_{l=0}^{\big{[}\nu/2\big{]}}D_{l_{1}+l_{2}-2l}P^{m_{1}+m_{2}}_{l_{1}+l_{2}-2l},$
(14)

where $\nu=l_{1}+l_{2}-m_{1}-m_{2}$ and $\big{[}\cdot\big{]}$ means the
integer part. In this way, for certain $l$ we may seek for all the products
$P^{m_{1}}_{l_{1}}P^{m_{2}}_{l_{2}}$ for which
$P^{m_{1}+m_{2}}_{l_{1}+l_{2}-2l}$ appears in the linear combination of Eq.
(14). We are thus searching for modes $(m_{i},l_{i})$, $i=1,2$, which “feed”
-by means of the products $P^{m_{1}}_{l_{1}}P^{m_{2}}_{l_{2}}$\- the mode
$(m,l)=(m_{1}+m_{2},l_{1}+l_{2}-2l)$. Some of the found modes $(m_{i},l_{i})$
are in resonance with the “feeded” mode $(m,l)=(m_{1}+m_{2},l_{1}+l_{2}-2l)$.

The linear combinations of Eq.(14) are tabulated in the appendix of Hwang
(1995), up to degree $5$, and help us to exemplify the relation between
resonances and nonlinear interaction of the modes. For example, by looking for
the products which feed $(m,l)=(3,3)$ we find

$\displaystyle
P^{2}_{2}P^{1}_{1}=g(P^{3}_{3}),~{}P^{2}_{3}P^{1}_{2}=g(P^{3}_{5},P^{3}_{3}),$
(15)

where $g$ is a linear function different for each product. One of these
relations provide the resonance condition $f^{3}_{3}=f^{1}_{1}+f^{2}_{2}$. We
note that we have excluded products such as

$\displaystyle
P^{1}_{3}P^{2}_{2}=g(P^{3}_{5},P^{3}_{3}),~{}P^{1}_{4}P^{2}_{3}=g(P^{3}_{7},P^{3}_{5},P^{3}_{3}),$

since they are equivalent to those of Eq.(15) in the sense that for the modes
$(1,1)$ and $(1,3)$ (resp. $(1,2)$ and $(1,4)$) the same frequency can be
selected $f^{1}_{1}=f^{1}_{3}$ (resp. $f^{1}_{2}=f^{1}_{4}$), as they both
represent the equatorially symmetric (resp. equatorially antisymmetric) part
of the $m=1$ mode. We have also not considered the coupling with the $m=0$
mode.

Figure 9: As figure 7 but for the $(m,l)=(4,4)$ component of the nonlinear
term. In this figure the modes $(m,l)=(2,3),(2,3)$ and $(m,l)=(1,2),(3,4)$ are
considered in the middle and bottom row. The solution is a 2T MRW with $m=1$
azimuthal symmetry at ${\rm Ha}=3.5$.

Figure 10: As figure 7 but for the $(m,l)=(4,5)$ component of the nonlinear
term. In this figure the modes $(m,l)=(2,2),(2,3)$, the modes
$(m,l)=(1,2),(3,3)$, and the modes $(m,l)=(1,1),(3,4)$, are considered in the
2nd, 3rd, and 4th rows, respectively. The solution is a 2T MRW with $m=1$
azimuthal symmetry at ${\rm Ha}=3.5$.

In case of the $(m,l)=(3,4)$ mode the situation is analogous and the products
feeding this mode are

$\displaystyle
P^{2}_{2}P^{1}_{2}=g(P^{3}_{4}),~{}P^{2}_{3}P^{1}_{1}=g(P^{3}_{4}),~{}P^{2}_{3}P^{1}_{1}=g(P^{3}_{4}),$

for which both resonance relations have been found (see Table 1) in the
scenario studied in § IV.2. For the azimuthal wave number $m=4$ the products
feeding the $(4,4)$ mode are

$\displaystyle
P^{2}_{2}P^{2}_{2}=g(P^{4}_{4}),~{}P^{2}_{3}P^{2}_{3}=g(P^{4}_{6},P^{4}_{4}),$
$\displaystyle
P^{3}_{3}P^{1}_{1}=g(P^{4}_{4}),~{}P^{3}_{4}P^{1}_{2}=g(P^{4}_{6},P^{4}_{4})$

and for the $(4,5)$ mode are

$\displaystyle
P^{2}_{3}P^{2}_{2}=g(P^{4}_{5}),~{}P^{3}_{3}P^{1}_{2}=g(P^{4}_{5}),~{}P^{3}_{4}P^{1}_{1}=f(P^{4}_{5}).$

As in the case for the $m=3$ wave number the resonances found for the $(4,4)$
and $(4,5)$ modes arise from the above relationships. We have also checked the
case $m=5$.

To further investigate the coupling between the resonant modes for the
poloidal field we compute its associated nonlinear term from the Navier-Stokes
equation, i. e.,

$\mathcal{N}({\mathbf{v}})={\rm
Re}~{}\mathbf{r}\cdot\nabla\times\nabla\times{\bf{\mathbf{w}}\times{\mathbf{v}}},$
(16)

but with only considering the poloidal component when computing the velocity
${\mathbf{v}}$, and vorticity ${\mathbf{w}}$ fields (i. e. we set the toroidal
field to zero $\Psi\equiv 0$), so $\mathcal{N}\equiv\mathcal{N}(\Phi)$. We
then compare the spatial patterns of the scalar $\mathcal{N}(\Phi)$ obtained
by using all the modes in the spherical harmonic expansion of the poloidal
field (Eq. (5)), with the patterns of $\mathcal{N}(\Phi)$ obtained from a
poloidal field with only nonzero amplitudes for a selected set of two resonant
modes. As it is shown in the following, the set of two resonant modes provides
a first order approximation of the patterns obtained with the full spherical
harmonic expansion. For the pattern comparison we have considered the 2T MRW
with $m=1$ azimuthal symmetry at ${\rm Ha}=3.5$, the resonances of which have
been studied in § IV.2, concretely in table 1.

The first row of figure 7 displays the contour plots of
$\big{[}\mathcal{N}(\Phi)\big{]}^{3}_{3}$, i. e., the spherical harmonic
amplitude $(m,l)=(3,3)$ of the nonlinear term given in Eq. 16, on spherical,
meridional and latitudinal sections (from left two right). The position of the
sections, marked with dashed lines, matches the position of a local maxima.
The second row shows the corresponding contour plots of
$\big{[}\mathcal{N}(\Phi)\big{]}^{3}_{3}$ for the case that only the
amplitudes $(m,l)=(1,1),(2,2)$ of the poloidal field are non-zero. The
similarity between the radial structure among the two rows is a clear
indication that the nonlinear interaction of the $(1,1)$ and $(2,2)$ resonant
modes is sufficient to provide a first order approximation of the nonlinear
term feeding the equatorially symmetric $(3,3)$ mode. Figure 8 summarises the
comparison in the case of the modes which are resonant with the equatorially
antisymmetric mode $(3,4)$ (shown in the first row). These are the modes
$(1,2),(2,2)$ and $(1,1),(2,3)$ displayed in the middle and bottom row,
respectively.

The patterns of the nonlinear terms in the case of the resonant modes with
azimuthal wave number $m=4$ are displayed in Fig. 9, for the equatorially
symmetric $(4,4)$ mode, and in Fig 10, for the equatorially antisymmetric
$(4,5)$ mode, respectively. The patterns for the resonant modes $(2,3),(2,3)$
and $(1,2),(3,4)$ and those obtained from the full spherical harmonics
expansion match only close to the inner boundary, so non-resonant modes should
be considered to approximate the nonlinear term close to the outer boundaries.
For the equatorially antisymmetric mode $(4,5)$ the resonant modes provide a
reasonable approximation of the nonlinear term.

##  VI Conclusions

A comprehensive frequency analysis of several equatorially symmetric and
antisymmetric poloidal modes has been presented for different types of weakly
magnetic spherical Couette flows, in the context of the radial jet instability
(Hollerbach, Junk, and Egbers (2006); Travnikov, Eckert, and Odenbach (2011);
Garcia and Stefani (2018)). Two different bifurcation scenarios have been
considered to evidence the mechanisms for the development of resonant triads
among modes $(m,l)$, having different spherical harmonic order $m$ and degree
$l$.

The first scenario comprises the first, second and third bifurcations from the
base state giving rise to stable flows. The second scenario involves up to
five bifurcations giving rise to either unstable or stable flows. In both
scenarios the sequence of bifurcations includes the path:

$\text{base
state}\rightarrow\text{RWs}\rightarrow\text{2T-MRWs}\rightarrow\text{3T-MRWs}$

for which the right arrows indicate Hopf bifurcations, which may break the
azimuthal symmetry of the flow. The above sequence of bifurcations is generic
in SO$(2)\times$Z2-equivariant systems when the first bifurcation breaks the
axisymmetry of the base state. In this case, RWs are the first time dependent
solutions (e. g. Rand (1982); Crawford and Knobloch (1991); Golubitsky,
LeBlanc, and Melbourne (2000)).

The type of time dependence of a RW, for which the main time scale is
associated to a solid body rotation of a flow pattern, allows to define
$\overline{\varphi}=\varphi+\omega t$, with $\omega$ being the rotation
frequency of the wave, so the mathematical description in terms of spherical
harmonics become equivalent to solutions of the inviscid equation (Greenspan
(1968)). This is also true for MRW which are quasiperiodic flows with two
incommensurate frequencies for which, due to the nonlinear coupling, resonance
relations can be defined following similar arguments as those given in Barik
_et al._ (2018).

For all the MRW explored, which have the azimuthal symmetry broken, a
multitude of resonance relations, both in terms of spherical harmonic order
$m$ and degree $l$, are found. We have considered modes of the type $(m,m)$
and $(m,m+1)$ to investigate the role of the equatorially symmetric and
antisymmetric part of the flow. In addition, we have analyzed the nonlinear
interactions giving rise to triadic resonances in terms of Legendre
polynomials.

The phase coupling among resonant modes, pointed out in Hoff, Harlander, and
Triana (2016) and further investigated in Barik  _et al._ (2018), is a
consequence of the regularity of MRW, as they are quasiperiodic solutions. For
this type of solutions, frequency spectra do not depend on time (Garcia  _et
al._ (2021)). In addition, triadic resonances are observed for chaotic flows
originated from further bifurcations of MRWs. This is because the main two
frequencies of these chaotic flows are very similar to those of the MRW, from
which they bifurcate (Garcia  _et al._ (2020a, b)). Because of this, triadic
resonances are very common for the radial jet instability of the MSC flow.

The results of the present study are compared with previous studies devoted to
triadic resonances in spherical systems. In the case of spherical Couette flow
resonant modes have been studied in Barik  _et al._ (2018) and for rotating
thermal convection in full spheres resonant solutions are investigated in Lin
(2021). In contrast to these studies, for which all the free resonant modes
are inertial and the base mode is strongly geostrophic, our simulations
demonstrate the existence of resonances between modes which are neither
geostrophic nor inertial, but related with the radial jet instability.

The existence of triadic resonances in spherical systems can thus be explained
in terms of bifurcation theory since they are associated to the existence of
MRW. In this framework, we argue that an inertial regime is not a requirement
for the development of resonant triads as our results point out. We remark
that in spherical systems the interpretation of resonances in terms of MRW, i.
e. equivariant dynamical systems theory, is not in contradiction with the
interpretation in terms of inertial waves since MRWs in the inertial regime
have been found in the case of rotating thermal convection (e. g. Garcia,
Chambers, and Watts (2019); Lin (2021)).

We finally note that MRW are common in rotating convective spherical shells
(e. g. Garcia  _et al._ (2015); Garcia, Net, and Sánchez (2016)) in a regime
in which the onset of convection is of columnar type (Zhang (1992)). We have
checked that for these type of MRW resonant triads also occur.

###### Acknowledgements.

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No 787544).

## Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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