11institutetext: Department of Earth and Space Sciences, Southern University
of Science and Technology, Shenzhen 518055, China
11email<EMAIL_ADDRESS>
# Dynamical tides in Jupiter and the role of interior structure

Yufeng Lin

###### Abstract

Context. The Juno spacecraft has obtained highly accurate tidal Love numbers,
which provide important constraints on the tidal response and interior
structure of Jupiter.

Aims. In order to exploit these observations, it is necessary to develop an
approach for accurately calculating the tidal response of Jupiter for a given
interior model and to investigate the role of interior structure.

Methods. We directly solve the linearized tidal equations of a compressible,
self-gravitating, rotating and viscous fluid body using a pseudo-spectral
method. The Coriolis force is fully taken into account but the centrifugal
effect is neglected. We can simultaneously obtain the real and imaginary parts
of the tidal Love numbers for a given planetary interior model.

Results. We calculate the tidal responses for three simple interior models of
Jupiter which may contain a compact rigid core or an extended dilute core. All
of models we consider can explain the fractional correction $\Delta
k_{22}\approx-4\%$ due to dynamical tides, but all have difficulties to
reconcile the observed $\Delta k_{42}\approx-11\%$ for the high-degree tidal
Love number. We show that the Coriolis force significantly modifies gravity
modes in an extended dilute core at the tidal frequency relevant to the
Galilean satellites. We demonstrate that a thin stable layer in the outer
region, if exists, would also influence the tidal responses of Jupiter.

###### Key Words.:

giant planets – tides – internal structure

##  1 Introduction

Tidal interactions between Jupiter and the Galilean satellites play an
important role in the orbital evolution of the system and the internal
dynamics of the moons (Lainey et al., 2009). The highly active volcanic
eruptions on Io are believed to be due to strong tides raised by Jupiter
(Peale et al., 1979). Meanwhile, tides are also raised in Jupiter by its
moons, probably dominated by Io (Gavrilov & Zharkov, 1977). The tidal response
of a gaseous body such as Jupiter is conventionally treated as a hydrostatic
deformation, which acquires a small phase lag with respect to the tidal
forcing due to dissipative processes. This is known as the equilibrium tide.
However, the equilibrium tide alone does not suffice to account for the
observed strong tidal dissipation in Jupiter (Lainey et al., 2009) and the
gravitational perturbations recently measured by the Juno spacecraft (Durante
et al., 2020).

In fact, the equilibrium tide does not satisfy the momentum equation of tidal
flows and thus corrections have to be made to fully account for the tidal
response of Jupiter. The corrections to the equilibrium tide are collectively
referred to as the dynamical tide, which usually involves wave-like motions in
the planet and depends on the tidal frequency and the interior structure
(Ogilvie, 2014). The dynamical tide may provide extra channels of tidal
dissipation and produce additional gravitational perturbations on top of the
hydrostatic deformation. The Juno spacecraft has obtained highly accurate
tidal Love numbers $k_{lm}$ (Durante et al., 2020), which quantitatively
characterize the tidal response of Jupiter to a tidal forcing component
represented in spherical harmonics of degree $l$ and order $m$. The observed
tidal Love numbers by Juno exhibit non-negligible discrepancies with respect
to the theoretically calculated hydrostatic values (Wahl et al., 2020),
suggesting that the dynamical tide has to be considered to explain the
observed tidal response. Specifically, Juno observations found $\Delta
k_{22}\approx-4\%$ for the dominant tidal component $l=2$ and $m=2$ and
$\Delta k_{42}\approx-11\%$ for the high-degree tidal component $l=4$ and
$m=2$, where $\Delta k_{lm}=(k_{lm}-k_{lm}^{(hs)})/k_{lm}^{(hs)}$ represents
the fractional correction to the hydrostatic value $k_{lm}^{(hs)}$ (Wahl et
al., 2020; Idini & Stevenson, 2021, 2022a).

As the dynamical tides is sensitive to the tidal frequency and the interior
structure, the detected gravitational signatures of dynamical tides may
provide important constraints on the Jupiter’s interior (Idini & Stevenson,
2021; Lai, 2021; Idini & Stevenson, 2021, 2022b; Dewberry & Lai, 2022). Recent
studies (Idini & Stevenson, 2021; Lai, 2021) have revealed that the
discrepancy in $k_{22}$ can be mainly attributed to the Coriolis effect on the
fundamental modes, i.e. $f-$modes. More recently, Idini & Stevenson (2022b)
proposed that the resonant locking with a gravity mode in an extended dilute
core can explain $\Delta k_{42}\approx-11\%$. This provides an independent
constraint on the existence of a dilute core in Jupiter, which has also been
suggested by the Juno measurements of gravitational moments of Jupiter (Wahl
et al., 2017; Militzer et al., 2022). However, the tidal constraint on the
existence of a dilute core remains some uncertainties. The calculation of
tidal response in Idini & Stevenson (2022b) inadequately treated the
rotational (Coriolis) effect, which plays an important role in Jupiter’s tidal
responses because the tidal frequencies of Galilean satellites are comparable
to the spin frequency of Jupiter. Including the Coriolis force introduces
inertial waves in the neutrally buoyant regions (Ogilvie & Lin, 2004; Wu,
2005a) and mixed gravity waves and inertial waves, i.e. gravito-inertial
waves, in the stably stratified region (Dintrans et al., 1999; Xu & Lai,
2017). The mechanism proposed by Idini & Stevenson (2022b) is also struggling
to reconcile both the real part (relevant to the gravitational perturbation)
and imaginary part (relevant to the tidal dissipation) of the tidal Love
numbers.

In this study, we develop a method to directly calculate the tidal response of
a fully compressible, self-gravitating, rotating and viscous fluid body. The
Coriolis force is fully taken into account but the centrifugal force is
neglected, which allows us to numerically solve the problem in spherical
geometry using a pseudo-spectral method based on spherical harmonic expansions
(Ogilvie & Lin, 2004; Lin & Ogilvie, 2017). As we directly solve the tidally
forced problem with explicit viscosity, we can simultaneously obtain the real
and imaginary parts of the tidal Love number for a given planetary interior
model. Our approach is different from recent studies on dynamical tides of
Jupiter (Lai, 2021; Idini & Stevenson, 2022b; Dewberry & Lai, 2022). They
obtain the eigen modes of the inviscid fluid body first and then calculate the
tidal Love number (only the real part) through projecting the tidal force onto
each eigen modes. We consider three nominal interior models of Jupiter to
investigate the dependence of the tidal response on the tidal frequency and
the interior structure. We focus on the effect of a compact rigid core, an
extended dilute core and a thin stably stratified layer in the outer region on
tidal responses. All of simplified models can explain the observed $\Delta
k_{22}\approx-4\%$ as previous studies have shown. However, these simplified
models are difficult to account for the observed $\Delta k_{42}\approx-11\%$.
Resonances with gravito-inertial modes in an extended dilute core near the
tidal frequency of Io can produce non-negligible dynamical correction to
$k_{42}$, but it is insufficient to explain the Juno observation based on our
simplified model.

##  2 Tidal model

We consider linear tidal responses of a rotating gaseous planet to a tidal
potential component of
$\Psi_{l}^{m}=\mathcal{A}(r/R)^{l}Y_{l}^{m}(\theta,\phi)\mathrm{e}^{-\mathrm{i}\omega
t}$, where $\mathcal{A}$ is the tidal amplitude, $R$ is the radius of the
planet, $Y_{l}^{m}(\theta,\phi)$ represents spherical harmonics and $\omega$
is the tidal frequency. The resulting tides of the planet produce an external
gravitational potential perturbation
$\Phi^{\prime}=\mathcal{B}(R/r)^{l+1}Y_{l}^{m}(\theta,\phi)\mathrm{e}^{-\mathrm{i}\omega
t}$ (and probably other spherical harmonic components). The ratio
$K_{l}^{m}(\omega)=\mathcal{B}/\mathcal{A}$ defines the tidal Love number,
which depends on the tidal frequency. The tidal Love number $K_{l}^{m}$ is a
complex number because there exists a phase lag between the forcing and the
gravitational perturbations due to dissipative processes (Ogilvie, 2014).
While the real part $k_{lm}=\mathrm{Re}[K_{l}^{m}]$ measures the in-phase
gravitational perturbations with the tidal forcing, the imaginary part
$\mathrm{Im}[K_{l}^{m}]$ quantifies the out of phase tidal response and is
related to the dissipation rate. The ratio between the real and imaginary
parts is related to the tidal quality factor

$Q=\mathrm{sgn}(\omega)\frac{k_{lm}}{\mathrm{Im}[K_{l}^{m}]},$ (1)

where $\mathrm{sgn}(\omega)=\pm 1$ is the sign function. Because the phase lag
is generally very small, i.e. $Q\gg 1$, the magnitude of the imaginary part is
typically much smaller than the real part. In this study, we develop an
approach to directly and simultaneously calculate the real and imaginary parts
of the tidal Love number for a given planetary model.

###  2.1 Linearized equations

For a compressible, self-gravitating and rotating fluid body which may contain
a rigid core of radius $R_{i}$, linear perturbations to a tidal potential
$\Psi\propto\mathrm{e}^{-\mathrm{i}\omega t}$ in the rotating frame are
described by the following equations (e.g. Ogilvie & Lin, 2004):

$-i\omega\bm{u}^{\prime}=-2\bm{\Omega\times
u^{\prime}}-\frac{1}{\rho_{0}}\bm{\nabla}P^{\prime}+\frac{\rho^{\prime}}{\rho_{0}^{2}}\bm{\nabla}P_{0}-\bm{\nabla}\Phi^{\prime}-\bm{\nabla}\Psi+\bm{f}_{\nu},$
(2) $-i\omega\rho^{\prime}+\bm{\nabla\cdot}(\rho_{0}\bm{u}^{\prime})=0,$ (3)

$-i\omega\left(\frac{P^{\prime}}{\Gamma
P_{0}}-\frac{\rho^{\prime}}{\rho_{0}}\right)+\bm{u}^{\prime}\cdot\left(\frac{1}{\Gamma}\nabla\ln
P_{0}-\nabla{\ln\rho_{0}}\right)=0$ (4) $\nabla^{2}\Phi^{\prime}=4\pi
G\rho^{\prime},$ (5)

where $\bm{u}$ is the velocity, $\bm{\Omega}$ the rotation rate, $\rho$ the
density, $P$ the pressure, $\Gamma$ the adiabatic index and $G$ the
gravitational constant. In the above equations, the subscript 0 denotes
physical quantities in hydrostatic state (without tidal potential) and the
notations with the prime represent Eulerian perturbations induce by the tidal
forcing. In the momentum equation (2), we explicitly include a viscous force
$\bm{f}_{\nu}$ defined as

$\bm{f}_{\nu}=\frac{1}{\rho_{0}}\bm{\nabla}\cdot(2\mu\bm{S}),$ (6)

where $\mu$ is the dynamic shear viscosity (we neglect the bulk viscosity) and
$\bm{S}$ is the strain-rate tensor:

$\bm{S}=\frac{1}{2}\left[\bm{\nabla}u^{\prime}+(\bm{\nabla}\bm{u}^{\prime})^{T}\right]-\frac{1}{3}(\bm{\nabla}\cdot\bm{u}^{\prime})\mathbf{I}.$
(7)

Note that we include the viscous force in the momentum equation but neglect
the viscous heating in the energy equation, i.e. the density and pressure
perturbations are treated as adiabatic.

In this study, we take fully into account the Coriolis force due to the
rotation but neglect the centrifugal distortion for numerical convenience. The
centrifugal effect can be measured by $\epsilon=\Omega/\omega_{dyn}$, i.e. the
ratio between the spin frequency $\Omega$ and the dynamical frequency
$\omega_{dyn}=(GM/R^{3})^{1/2}$, which is not particularly small for Jupiter
($\epsilon=0.288$). Indeed, the centrifugal distortion of Jupiter has non-
negligible contributions to the total Love number $k_{lm}$, especially for the
high-degree Love number $k_{42}$ because the tidal response at $l=m=2$ can
produce a gravitational perturbation at $l=4$ and $m=2$ in an oblate figure
(Idini & Stevenson, 2022a). For the hydrostatic $k_{42}^{(hs)}$ of Jupiter due
to Io, 93% of the total value is actually contributed by the centrifugal
coupling with $k_{22}$ and only the remaining 7% is produced by the tidal
forcing at $l=4$ and $m=2$ (Wahl et al., 2020; Idini & Stevenson, 2022a). In
this paper we do not aim to directly fit the $k_{lm}$ observed by Juno, but
rather focus on the fractional corrections $\Delta k_{lm}$ by the dynamical
tides. In terms of the fractional correction $\Delta k_{lm}$, the centrifugal
contribution to $\Delta k_{22}$ can be neglected in leading order (Lai, 2021).
However, the centrifugal contribution to $\Delta k_{42}$ can not be neglected
even in leading order because the $k_{42}^{(hs)}$ is mostly contributed by the
centrifugal coupling with $k_{22}$. This complicates the comparison between
the calculated $\Delta k_{42}$ in a spherical figure and the observation.
Nevertheless, the calculated $\Delta k_{42}$ in a spherical figure can be
multiplied by the factor 0.07 to account Jupiter’s centrifugal coupling effect
for qualitative comparisons with the observation (Idini & Stevenson, 2022b).
Such a comparison would assume that the tidally-excited internal modes are not
significantly modified by the centrifugal deformation.

By neglecting the centrifugal deformation, the unperturbed basic state is
spherically symmetric, i.e. depends on the radius $r$ only. Given the density
$\rho_{0}(r)$ and pressure $P_{0}(r)$ profiles of the unperturbed state, the
radial gravitational acceleration (inward) $g(r)$ and the Brunt-Väisälä
frequency $N(r)$ are then determined by

$g=\frac{d\Phi_{0}}{dr}=-\frac{1}{\rho_{0}}\frac{dP_{0}}{dr},$ (8)
$N^{2}=g\left(\frac{1}{\Gamma}\frac{d\ln
P_{0}}{dr}-\frac{d\ln\rho_{0}}{dr}\right).$ (9)

###  2.2 Numerical method

In order to obtain the complex Love numbers, we numerically solve Eqs. (2-5)
using a pseudo-spectral method for the prescribed basic states, subject to the
relevant boundary conditions. The numerical scheme is based on the method used
in previous studies (Ogilvie & Lin, 2004; Lin & Ogilvie, 2017), but we extend
the method to solve the full set of linearized equations (2-5) without making
a low-frequency approximation (Ogilvie, 2013). By introducing
$h^{\prime}=P^{\prime}/\rho_{0}$ and eliminating the density perturbation
$\rho^{\prime}$, Eqs. (2-5) can be reduced to the following equations:

$\displaystyle-i\omega\rho_{0}\bm{u}^{\prime}$ $\displaystyle=$
$\displaystyle-2\rho_{0}\bm{\Omega\times
u^{\prime}}-\bm{\nabla}(\rho_{0}h^{\prime})+\bm{g}\nabla^{2}\varphi^{\prime}/(4\pi
G)$ (10)
$\displaystyle-\rho_{0}\bm{\nabla}\varphi^{\prime}-\bm{\nabla}\Psi+\nabla\cdot(2\mu\bm{S}),$

$-i\omega
h^{\prime}=-c_{s}^{2}(N^{2}u^{\prime}_{r}/g+\bm{\nabla\cdot}(\rho_{0}\bm{u}^{\prime})/\rho_{0}),$
(11)

$-i\omega\nabla^{2}\varphi^{\prime}=-4\pi
G\bm{\nabla\cdot}(\rho_{0}\bm{u}^{\prime}),$ (12)

where $u_{r}^{\prime}$ is the radial velocity perturbation and
$c_{s}^{2}=\Gamma P_{0}/\rho_{0}$ is the square of the adiabatic sound speed.

We impose boundary conditions including the regularity of the gravitational
perturbations, zero radial velocity on the rigid inner boundary and vanishing
Lagrange pressure perturbation at the surface, i.e. $\delta
P=P^{\prime}+u_{r}^{\prime}/(-i\omega)\nabla P_{0}=0$. In terms of
$h^{\prime}$ and $u_{r}^{\prime}$, the last boundary condition can be written
as (Dewberry et al., 2021)

$\left(-i\omega h^{\prime}-gu_{r}^{\prime}\right)|_{r=R}=0.$ (13)

As the viscous force is included, additional boundary conditions are required
to complete the boundary value problem. We use the so-called stress-free
conditions, i.e. the tangential stresses vanish, at both boundaries.

For a given tidal potential
$\Psi_{l}^{m}=\mathcal{A}(r/R)^{l}Y_{l}^{m}(\theta,\phi)\mathrm{e}^{-\mathrm{i}\omega
t}$, the tidal perturbations (including both equilibrium and dynamical tides)
$\bm{u}^{\prime}$, $h^{\prime}$ and $\Phi^{\prime}$ can be expanded as

$\bm{u}^{\prime}=\sum_{n=m}^{L}u_{n}^{m}(r)\bm{R}_{n}^{m}+\sum_{n=m}^{L}v_{n}^{m}(r)\bm{S}_{n}^{m}+\sum_{n=m}^{L}w_{n}^{m}(r)\bm{T}_{n}^{m},$
(14) $h^{\prime}=\sum_{n=m}^{L}h_{n}^{m}(r)Y_{n}^{m}(\theta,\phi),$ (15)
$\Phi^{\prime}=\sum_{n=m}^{L}\Phi_{n}^{m}(r)Y_{n}^{m}(\theta,\phi),$ (16)

where $\bm{R}_{n}^{m}$, $\bm{S}_{n}^{m}$, $\bm{T}_{n}^{m}$ are vector
spherical harmonics

$\bm{R}_{n}^{m}=Y_{n}^{m}(\theta,\phi)\bm{\hat{r}},\quad\bm{S}_{n}^{m}=r\bm{\nabla}Y_{n}^{m}(\theta,\phi),\quad\bm{T}_{n}^{m}=r\bm{\nabla\times}\bm{R}_{n}^{m}.$
(17)

As the basic state is axisymmetric, the perturbations involve spherical
harmonics with the same order $m$ as the tidal potential $\Psi_{l}^{m}$, but
the Coriolis force would couple all spherical harmonics with degree $n\geq m$.
For numerical calculations, we have to make a truncation at certain degree
$L$. Substituting expansions of Eqs. (14-16) into Eqs. (10-12) and projecting
onto spherical harmonics, we end up with a set of ordinary differential
equations involving $u_{n}^{m}(r)$, $v_{n}^{m}(r)$, $w_{n}^{m}(r)$,
$h_{n}^{m}(r)$ and $\Phi_{n}^{m}(r)$ . For the radial dependence, we use
Chebyshev collocation on $N_{r}$ Gauss–Lobatto nodes (Rieutord et al., 2001).
The boundary conditions are applied through replacing the ODEs with the
corresponding boundary conditions on the boundary nodes. The regularity of
gravitational perturbations requires

$r\frac{d\Phi_{n}^{m}}{dr}+(n+1)\Phi_{n}^{m}=0\quad\mathrm{at}\ r=R,$ (18)
$r\frac{d\Phi_{n}^{m}}{dr}-n\Phi_{n}^{m}=0\quad\mathrm{at}\ r=R_{i}.$ (19)

The vanishing Lagrangian pressure perturbation at the surface and zero radial
velocity at the rigid inner boundary give

$-i\omega h_{n}^{m}=gu_{n}^{m}\quad\mathrm{at}\ r=R,$ (20)
$u_{n}^{m}=0\quad\mathrm{at}\ r=R_{i}.$ (21)

The stress-free boundary condition is given as (Ogilvie, 2009)

$u_{n}^{m}+r\frac{dv_{n}^{m}}{dr}-v_{n}^{m}=0,\quad
r\frac{dw_{n}^{m}}{dr}-w_{n}^{m}=0,$ (22)

at both boundaries.

Using the numerical discretization described above, the boundary value problem
becomes a linear system involving a large complex block-tridiagonal matrix.
The solution of the linear system is obtained using the standard direct
solver. We use typical truncations of $L=200$ and $N_{r}=100$ in this study.

Once the solution of the linear system is obtained numerically, the complex
tidal Love number is readily given by

$K_{l}^{m}=\Phi_{l}^{m}(r=R),$ (23)

for the tidal potential component
$\Psi_{l}^{m}=\mathcal{A}(r/R)^{l}Y_{l}^{m}(\theta,\phi)\mathrm{e}^{-\mathrm{i}\omega
t}$ (we simply set $\mathcal{A}=1$ for the linear tidal response). Note that
the solution includes both the equilibrium and dynamical tides. For the real
part of Love numbers, of particular interest is the fractional correction of
dynamical tides

$\Delta k_{lm}=(k_{lm}-k_{lm}^{(hs)})/k_{lm}^{(hs)},$ (24)

where $k_{lm}^{(hs)}$ is the hydrostatic value and is calculated by setting
$\omega=0$. As our calculations neglect the centrifugal effect which
significantly influences the high-degree Love number $k_{42}$, the calculated
value of $\Delta k_{42}$ should be multiplied by the factor 0.07 when compared
with the observation as we have discussed in Sec. 2.1.

We can also calculate the tidal dissipation rate $D_{\nu}$ from the velocity
perturbations

$D_{\nu}=\int_{V}2\mu\bm{S}^{2}dV,$ (25)

where the integral is taken over the fluid domain. The dissipation rate is
related to the imaginary part of the tidal Love number (Ogilvie, 2014)

$D_{\nu}=\frac{(2l+1)R\mathcal{A}^{2}}{8\pi G}\omega\mathrm{Im}[K_{l}^{m}],$
(26)

which can be served as an independent validation of the numerical code. The
above relation is satisfied to a high degree of accuracy in all of numerical
calculations presented in this paper.

###  2.3 Interior models

(a) (b) (c)

Figure 1: Three nominal models of Jupiter’s interior used in this study. The
top panel shows the schematic models and the bottom panel shows the density
(normalized by the density at the center) and the Brunt-Väisälä frequency
(normalized by the dynamical frequency) as a function of the radius. The blue
shadow in the bottom panel indicates solid regions. (a) A compact rigid core
model; (b) an extended dilute core model; (c) a compact rigid core and an
outer stable layer model.

In order to solve Eqs. (10-12), we need to prescribe basic state profiles
$\rho_{0}(r)$, $g(r)$ and $N^{2}(r)$ to model Jupiter’s interior. Our
understanding of Jupiter’s interior has been significantly improved by Juno
observations (Stevenson, 2020), yet it remains some degrees of uncertainties.
In this study, we do not aim to build a realistic model of Jupiter’s interior,
but focus on the fractional contributions of dynamical tides to the tidal Love
number for different possible scenarios of Jupiter’s interior. We consider
three nominal interior models (Fig. 1) based on a polytrope of index 1, which
is a good leading order approximation for Jupiter (Stevenson, 2020).

For all of models used in this study, the unperturbed density and gravity
follow a hydrostatic polytrope of index 1,

$\rho_{0}=\frac{\pi M}{4R^{3}}\frac{\sin kr}{kr},$ (27)
$g=\frac{GM}{r^{2}}\left[\sin(kr)-kr\cos(kr)\right],$ (28)

where $k=\pi/R$. The first model consists of a small rigid core of radius
$0.25R$ and an isentropic fluid envelope, i.e. $\Gamma=2$ and $N^{2}=0$ in the
fluid region (Fig. 1(a)).

The second model assumes an extended dilute core of radius $0.7R$ and an
isentropic envelope (Fig. 1(b)).The dilute core is treated as a stably
stratified fluid layer with the Brunt-Väisälä frequency given by

$\frac{N^{2}}{\omega_{dyn}^{2}}=\tilde{N}_{2}\sin\left(\frac{\pi
r}{R_{c}}\right),$ (29)

where $\tilde{N}_{2}=0.25$ and $R_{c}=0.7$ for this model. As we fixed the
density and pressure profiles to that of a polytrope, the stratification is
effectively realized by adjusting the adiabatic index $(\Gamma>2)$ in the
dilute core (Lai, 2021). This model is similar to that used in Idini &
Stevenson (2022b), but they adjust the density profile to model the stable
stratification in the dilute core while fix the adiabatic index $\Gamma=2$.

The third model is based on the model in Fig. 1(a), but we further add a
stably stratified layer between $0.8R$ and $0.9R$ (Fig. 1(c)), possibly
resulting from H-He immiscibility (Debras & Chabrier, 2019; Stevenson et al.,
2022). The Brunt-Väisälä frequency in the top stable layer is prescribed as

$\frac{N^{2}}{\omega_{dyn}^{2}}=\tilde{N}_{2}\frac{1}{[1+\mathrm{e}^{-100(r-0.8)}][1+\mathrm{e}^{100(r-0.9)}]}.$
(30)

The degree of stratification of this layer remains uncertain, but it is
estimated that typical values of $N^{2}/{\omega_{dyn}^{2}}$ would be roughly
between 0.1 and 0.8 for Jupiter (Christensen et al., 2020; Gastine & Wicht,
2021). Here we set a moderate value $\tilde{N}_{2}=0.5$.

Note that an interior model with the co-existence of a dilute core and a top
stable layer is also possible (Debras & Chabrier, 2019). As this kind of model
involves two different stably stratified layers, it would be difficult to
characterize the role of the top stable layer on tides. We consider only the
combination of a compact rigid core and a top stable layer for simplicity.

In all of these models, we set the total mass $M$, the radius $R$ and the spin
rate $\Omega$ such that the ratio $\epsilon=\Omega/\sqrt{GM/R^{3}}=0.288$,
corresponding the value of Jupiter. Our calculations also require the fluid
viscosity, which is difficult to estimate in detail for giant planets. We
simply assume the dynamic viscosity $\mu$ is proportional to the background
density $\rho_{0}$, so the kinematic viscosity $\nu=\mu/\rho_{0}$ is constant.
The viscosity can be measured by the dimensionless number $Ek=\nu/(\Omega
R^{2})$, known as the Ekman number. We set $Ek=10^{-6}$ for most of
calculations (unless otherwise specified), roughly corresponding to the
effective viscosity based on mixing-length theory (Guillot et al., 2004).

As we have mentioned that we do not aim to construct a realistic interior
model for Jupiter in this study. These simplified models are designed to
investigate the effects of a compact rigid core, an extended dilute core and a
top stable layer on the tidal responses of Jupiter respectively. Nevertheless,
the fractional corrections $\Delta k_{lm}$ and the tidal quality factor $Q$
for these simplified models can be used to make some qualitative comparisons
with the observations (Lai, 2021; Idini & Stevenson, 2021, 2022b).

##  3 Results

In this paper, we focus on the dominant tidal component $\Psi_{2}^{2}$ and a
high-degree tesseral component $\Psi_{4}^{2}$, for which non-negligible
dynamical corrections have been detected as we have discussed in Sec. 1. Our
calculations are limited to the frequency range of
$-2\leq\omega/\Omega\leq-1$, relevant to the tidal frequencies of the Galilean
moons. Note that the negative tidal frequency means that the tidal forcing is
retrograde in the co-rotating frame with the planet based on our convention.
For the real part of Love numbers, we show the fractional correction $\Delta
k_{lm}$. In order to make comparisons with the Juno observation, the
calculated $\Delta k_{42}$ is multiplied by 0.07 to compensate the centrifugal
effect which is neglected in our calculations. Because of the negative tidal
frequency, the imaginary part of Love numbers is also negative in our
calculations and is related to the tidal quality factor by
$k_{lm}/Q_{l}=-\mathrm{Im}[K_{l}^{m}]$ according to Eq. (1).

###  3.1 Full polytrope model

Figure 2: Complex Love number as a function of the tidal frequency for a full
isentropic polytrope of index 1. Top panel shows the fractional correction
$\Delta k_{lm}$ of the real part of the Love numbers. The fractional
correction $\Delta k_{42}$ (orange curve in the top panel) is multiplied by
0.07. The bottom panel shows the minus imaginary part
$-\mathrm{Im}[K_{l}^{m}]$, which is equivalent to $k_{lm}/Q_{l}$. Vertical
dashed lines indicate tidal frequencies of four Galilean Moons of Jupiter
(from right to left: Io, Europa, Ganymede, Callisto). The horizontal dashed
line in the bottom panel represents the astrometric observation of the
frequency independent $k_{2}/Q_{2}$ from Lainey et al. (2009).

(a) (b)

Figure 3: Density perturbations (left half) and radial velocity perturbations
(right half) in the meridional plane to the tidal component $\Psi_{4}^{2}$ at
two resonant frequencies in Fig. 2. Amplitudes are normalized by the maximum
absolute values.

Before presenting results for the interior models in Fig. 1, we first show the
tidal response of a full isentropic polytrope, i.e. neutrally buoyant in the
whole fluid sphere. This model serves as a reference for other models and has
been used to investigate the dynamical tides of Jupiter in recent analytical
studies (Idini & Stevenson, 2021; Lai, 2021). Fig. 2 shows both the real and
imaginary parts of the Love numbers as a function of the tidal frequency for
the full polytrope model. We can see that $\Delta k_{22}$ is negative in the
frequency range we considered and smoothly varies as the tidal frequency
except a burst around $\omega/\Omega=-1.08$, which corresponds to a resonance
with an inertial mode. Away from resonances, our numerical results are
consistent with recent theoretical calculations and produce $\Delta
k_{22}\approx-4\%$ at the tidal frequency of Io (Lai, 2021; Idini & Stevenson,
2021). These studies also revealed that the dynamical correction $\Delta
k_{22}$ can be attributed to the Coriolis effect on the $f$-modes. Apart from
the $f$-modes, the rotating sphere of isentropic fluid also supports smooth
inertial modes restored by the Coriolis force in the frequency range of
$0<|\omega/\Omega|<2$ (Greenspan, 1968; Lockitch & Friedman, 1999). The burst
of $\Delta k_{22}$ at $\omega/\Omega=-1.08$ indeed is due to the resonant
excitation of the inertial mode as shown in Fig. 3(a), but we notice that the
resonance occurs only in a very narrow frequency range.

However, this inertial mode has more significant contributions to $\Delta
k_{42}$. The angular structure of an inertial mode cannot be described by a
single spherical harmonics in general (Lockitch & Friedman, 1999), but the
density perturbations (and thus the gravitational perturbations) are dominated
by the spherical harmonics $Y_{4}^{2}(\theta,\phi)$ for the resonant inertial
mode at $\omega/\Omega=-1.0836$ as we can see from Fig. 3(a). This suggests a
likely strong coupling between the tidal potential component $\Psi_{4}^{2}$
and the inertial mode in Fig. 3(a), i.e. large tidal overlap as described in
Wu (2005b), leading to significant dynamical corrections to $k_{42}$. The
dynamical correction can reach $\Delta k_{42}\approx-10\%$ (after the
centrifugal correction) near the resonance at $\omega/\Omega=-1.0836$.
However, the tidal frequencies of the Galilean satellites are too far away
from this resonance.

The curve of $\Delta k_{42}$ also shows a spike around $\omega/\Omega=-1.51$,
corresponding to a narrow resonance with a high degree inertial mode
($\rho^{\prime}$ is dominated by $Y_{6}^{2}(\theta,\phi)$ as shown in Fig.
3(b)). Interestingly, the tidal frequency of Io is close to this resonance,
but the dynamical correction caused by this resonant mode is insufficient to
account for the observed $\Delta k_{42}\approx-11\%$. The frequencies of
inertial modes in Fig. 3 are slightly shifted comparing to that calculated by
Lockitch & Friedman (1999) for a polytrope of index 1 (see their table 6 and
note different conventions for the sign of frequencies) because they assumed
$\epsilon\to 0$ whereas we set $\epsilon=0.288$.

The imaginary parts of the Love numbers in Fig. 2 show that resonances with
inertial modes significantly enhance the tidal dissipation. The enhanced
dissipation due to resonant inertial modes in a neutrally buoyant sphere has
been demonstrated by Wu (2005b) but using different density profiles. When the
tidal frequency is away from resonances, the dissipation rate for the full
isentropic polytrope is too small to account for the observed tidal quality
factor $Q$ (Lainey et al., 2009).

###  3.2 Compact rigid core model

Figure 4: As for Fig. 2 but for the interior model with a compact rigid core.
The fractional correction $\Delta k_{42}$ (orange curve in the top panel) is
multiplied by 0.07.

(a) (b)

Figure 5: Density perturbations (left half) and gravitational perturbations
(right half) in the meridional plane to the tidal component $\Psi_{4}^{2}$ for
the interior model with a compact rigid core at (a) $\omega/\Omega=-1.5092$
(resonance) and (b) $\omega/\Omega=1.53$ (non-resonance) with $Ek=10^{-7}$.
Amplitudes are normalized by the maximum absolute values.

We now consider tidal responses for the interior model with a compact rigid
core. Basically, the inner region ($r\leq 0.25R$) of a whole fluid polytrope
becomes solid for this model. Fig. 4 shows the frequency-dependence of the
Love numbers for the compact rigid core model. We can see that the real parts
are largely similar to that of a full polytrope, but the imaginary parts are
rather different from that of a full polytrope, showing enhanced tidal
dissipation by introducing the rigid core. The rigid core model also supports
inertial waves in the fluid envelope, but these waves have some peculiar
behaviors due to the singularity in a spherical shell (Stewartson & Rickard,
1969). Smooth inertial modes do not exist generally in a spherical shell even
with uniform density (Rieutord et al., 2001), and localized wave beams spawned
from the critical latitudes propagate in the bulk along the characteristics of
the inertial wave equations (e.g. Ogilvie, 2009). However, Lin & Ogilvie
(2021) recently revealed that resonant tidal responses in a spherical shell
correspond to eigen modes with large scale flows hidden beneath localized wave
beams using a uniform density model. Furthermore, it was shown that the hidden
large scale structures basically resemble inertial modes in a full sphere.
This is in line with our results for the non-uniform density model in this
study. The real parts $k_{22}$ and $k_{42}$ are relevant to only large scale
density perturbations, which are similar to inertial modes in a full sphere as
one can see from Fig. 5. Therefore, the curves of $\Delta k_{lm}$ for the
rigid core model resemble that of a full polytrope, but note slight shifts of
the resonant frequencies due to the presence of rigid core. As for the full
polytrope, the compact rigid core model can produce $\Delta k_{22}=-4\%$ as
observed, but it cannot produce sufficient dynamical correction in the high-
degree Love number $k_{42}$ near the tidal frequency of Io to account for the
observed $\Delta k_{42}=-11\%$.

On the other hand, the imaginary parts are largely modified by the presence of
small rigid core. We can see that the tidal dissipation is significantly
enhanced by the localized wave beams spawned from the critical latitudes both
in and out of resonances. The velocity perturbations in Fig. 5(b) indeed
exhibit localized waves propagating in the bulk, which can generate
significant viscous dissipation but do not produce much density and
gravitational perturbations. In Fig. 4, we also see that several peaks in the
tidal dissipation (bottom panel) do not lead to obvious fluctuations in
$\Delta k_{lm}$ (top panel), corresponding to resonances with higher degree
modes that have little contributions to the low degree ( i.e. $l=2$ and $l=4$)
gravitational perturbations.

In summary for the compact rigid core model, the tidal dissipation is
significantly enhanced with respect to the full polytrope case. This is in
line with the early work of Ogilvie & Lin (2004), who have shown the enhanced
tidal dissipation due to inertial waves in the convective envelope of rotating
stars and planets. The averaged dissipation in the tidal frequency range of
Galilean moons gives rise to comparable tidal quality factor as observed
(Lainey et al., 2009). However, the fractional correction to the real part of
Love number $\Delta k_{42}$ is insufficient to explain the observation.

###  3.3 Dilute core model

Figure 6: As for Fig. 2 but for the interior model with an extended dilute
core. Green lines represent results without including the Coriolis force. The
fractional correction $\Delta k_{42}$ (orange and green curves in the top
panel) is multiplied by 0.07.

(a) (b)

Figure 7: Density perturbations (left half) and radial velocity perturbations
(right half) in the meridional plane to the tidal component $\Psi_{4}^{2}$ for
the interior model with an extended dilute core. (a) Without including the
Coriolis force at $\omega/\Omega=-1.5193$ (resonance); (b) including the
Coriolis force at $\omega/\Omega=-1.4448$ (resonance). Amplitudes are
normalized by the maximum absolute values.

An extended dilute core rather than a compact core in Jupiter has been
suggested recently based on Juno gravitational measurements (Wahl et al.,
2017; Militzer et al., 2022). In this subsection, we consider tidal responses
for the interior model with an extended dilute core as shown in Fig. 1(b). The
dilute core is treated as a stably stratified layer which supports gravity
waves restored by the buoyancy. If the Coriolis force is fully taken into
account, dynamical tides in the dilute core region would be in the the form of
mixed gravity waves and inertial waves, i.e. gravito-inertial waves (Dintrans
et al., 1999; Xu & Lai, 2017). Idini & Stevenson (2022b) recently calculated
the tidal response of Jupiter with an extended dilute core, but they did not
fully consider the Coriolis effect, which turns out to important as we will
show.

Fig. 6 shows the frequency-dependence of the Love numbers for the dilute core
model. For the tidal component $\Psi_{2}^{2}$ (bule curves), the dynamical
correction $\Delta k_{22}$ is generally similar to that of the full polytrope
except the absence of obvious spikes for the dilute core model. However, the
imaginary part exhibits several peaks and troughs, suggesting possible
resonances with high degree mixed modes that enhance the tidal dissipation but
do not significantly contribute to the $l=2$ gravitational perturbations. The
overall tidal dissipation is also enhanced with respect to the full polytrope
due to the excitation of gravito-inertial waves in the dilute core and
inertial waves in the convective envelope. The frequency-averaged tidal
dissipation tends to be compatible with the observed tidal quality factor as
we can see from Fig. 6.

For the tidal component $\Psi_{4}^{2}$, Fig. 6 also shows results without
including the Coriolis force (green curves) for comparison. We note that the
fractional correction $\Delta k_{42}$ is always positive when the Coriolis
force is neglected, probably because the pure gravity modes enhance the in-
phase gravitational perturbations and thus produce positive dynamical
corrections. Nevertheless, we observe distinct resonant responses at certain
tidal frequencies from both real and imaginary parts of the Love number for
the non-Coriolis case. For instance, the resonance at $\omega/\Omega=-1.5193$,
which is close to the tidal frequency of Io, corresponds to the first gravity
mode of $l=4$ and $m=2$ as shown in Fig. 7 (a). Indeed, Idini & Stevenson
(2022b) proposed the resonant locking between this gravity mode 111They used
slightly different background density $\rho_{0}(r)$ and Brunt-Väisälä
frequency $N(r)$, so the mode frequency is slightly shifted. (referred to as
${}_{4}^{2}g_{1}$) and the Jupiter-Io orbital evolution to explain the
observed $\Delta k_{42}$ for Jupiter. In Idini & Stevenson (2022b), the
Coriolis force is neglected for the calculation of gravity modes, but
approximated rotational corrections are made to obtain the Love number.
However, taking fully into account the the Coriolis force significantly alter
the tidal responses as we can see from Fig. 6. The dynamical correction
$\Delta k_{42}$ exhibits several large fluctuations especially in the
frequency range of $-1.5<\omega/\Omega<-1$. This is due to the mixing of
gravity modes and inertial modes in the dilute core, leading to more chances
for resonances. The most significant dynamical corrections are produced near
the tidal frequency $\omega/\Omega=-1.2$, which is close to the frequency of
the purely inertial mode as shown in Fig. 3(a). Of course, the inertial mode
is mixed with gravity modes in the dilute core for this model. The resonance
around $\omega/\Omega=-1.2$ can produce more than $-10\%$ dynamical
corrections in $k_{42}$ (after the centrifugal correction), but it is too far
away from the tidal frequency of Io. The resonance close to the tidal
frequency of Io (also close to the frequency of pure gravity mode
${}_{4}^{2}g_{1}$) occurs at $\omega/\Omega=-1.4448$ when the Coriolis force
is considered. Fig. 7 (b) shows the spatial structure of this resonant
response. The Coriolis effect not only leads to a non-negligible shift in the
mode frequency, but also largely modifies the mode structure. The
perturbations are in the from of gravito-inertial waves in the dilute core and
become pure inertial waves in the neutrally buoyant envelope. Non-negligible
dynamical corrections are induced by this resonance at
$\omega/\Omega=-1.4448$, but the corrections are insufficient (after the
centrifugal correction) to account for the observed $\Delta k_{42}=-11\%$. As
the resonance is very narrow, we use 200 equally spaced frequency points in
the tidal frequency interval of [-1.45, -1.43]. The peak amplitude of $\Delta
k_{42}$ in this frequency interval is comparable to that of using only 20
frequency points, suggesting that the frequency sampling points are sufficient
to capture the resonant peak.

Comparing the orange and green curves in the bottom panel of Fig. 6, we can
see that the tidal dissipation is increased by about two orders of magnitude
when the Coriolis force is included. This suggests that the excitation of pure
gravity waves is a less efficient tidal dissipation mechanism (unless
resonances take place) based on our linear calculations, though the nonlinear
interaction or wave breaking of gravity waves may lead to efficient tidal
dissipation (e.g. Barker, 2011; Weinberg et al., 2012).

###  3.4 Outer stable layer model

Figure 8: As for Fig. 2 but for the interior model with a small rigid core and
a top stably stratified layer. Green lines represent results at the Ekman
number $Ek=10^{-7}$. The fractional correction $\Delta k_{42}$ (orange and
green curves in the top panel) is multiplied by 0.07.

(a) (b)

Figure 9: Perturbations in the meridional plane to the tidal component
$\Psi_{4}^{2}$ at $\omega/\Omega=-1.1650$ for the interior model (c) in Fig.
1. (a) Density (left half) and radial velocity (right half) perturbations; (b)
gravitational (left half) and vorticity (right half) perturbations. Amplitudes
are normalized by the maximum absolute values. The dashed lines denote
$r=0.8R$.

We finally consider the effect of an outer stable layer, which may exist in
Jupiter resulting from H-He immiscibility (Debras & Chabrier, 2019). Fig. 8
shows the Love numbers as a function of the tidal frequency for the interior
model (c) in Fig. 1, which includes a compact rigid core and a top stable
layer between $0.8R$ and $0.9R$. For the tidal responses to $\Psi_{2}^{2}$,
the dynamical correction $\Delta k_{22}$ is similar to the case without the
stable layer, but the presence of the thin stable layer eliminates the spike
due to the resonant inertial mode at the tidal frequency around
$\omega/\Omega=-1.08$. The overall tidal dissipation due to $\Psi_{2}^{2}$ is
comparable to the counterpart without the top stable layer (blue curve in the
bottom panel of Fig. 4), but the fluctuation amplitudes, i.e. the differences
between peaks and troughs, are smaller.

For the tidal responses to $\Psi_{4}^{2}$, we also show results for
$Ek=10^{-7}$ (green curves) to illustrate the effect of fluid viscosity in
Fig. 8. One can see that the viscosity has little influence on the real part
of the Love number. The tidal dissipation weakly depends on viscosity at peaks
and troughs, but the overall dissipation tends to be insensitive to viscosity.
Indeed, Ogilvie (2013) has shown the frequency-averaged dissipation is
independent of viscosity.

The dynamical correction $\Delta k_{42}$ is also similar to the case without
the stable layer. We can see large variations of $\Delta k_{42}$ at the tidal
frequency around $\omega/\Omega=-1.165$, which corresponds to a resonant mode
as shown in Fig. 9. This mode is complicated because it involves three
different layers for the interior model considered here. The fluid body is
primarily neutrally buoyant and supports inertial waves. However, the fluid
domain is separated by the thin stable layer, which suppresses radial fluid
motions and creates a ”barrier” for the communication between inertial waves
in the inner and outer regions (see the radial velocity and vorticity
perturbations in Fig. 9). In addition, the thin stable layer supports
rotationally modified gravity waves. The density perturbations are mainly
restricted in the stable layer and the outer envelope, i.e. in the region of
$r>0.8R$. Despite the complicated velocity and density perturbations, the
gravitational perturbations are dominated by the $l=4$ component with
relatively simple radial dependence. In this regard, this complicated mode is
relevant to the $l=4$ inertial mode without the stable layer, leading to large
dynamical corrections around the tidal frequency at $\omega/\Omega\approx-1.1$
as in Fig. 4. However, the dynamical correction $\Delta k_{42}$ is negligible
after the centrifugal correction at the tidal frequency of Io.

##  4 Conclusions

We have developed a numerical method for calculating the tidal responses of a
compressible, self-gravitating, rotating and viscous fluid body. We take fully
into account the Coriolis force but neglect the centrifugal distortion, which
allows us to solve the problem in the spherical geometry. We use the pseudo-
spectral method based on spherical harmonics in the angular directions and
Chebyshev collocation in the radial direction. Different from recent studies
on Jupiter’s dynamical tides (Lai, 2021; Idini & Stevenson, 2022b; Dewberry &
Lai, 2022), we directly solve the tidally forced problem and explicitly add
the fluid viscosity, which allows us to simultaneously obtain the real and
imaginary parts of the tidal Love numbers for a given planetary interior
model.

In this study, we considered three simplified interior models (Fig. 1) of
Jupiter based on a polytrope of index 1. We focus on the tidal components
$\Psi_{2}^{2}$ and $\Psi_{4}^{2}$ in the frequency range of
$-2\leq\omega/\Omega\leq-1$, which is relevant to the tidal frequencies of
Galilean moons. Our numerical results show that the dynamical correction
$\Delta k_{22}$ is generally insensitive to the interior models. All of models
we considered can give rise to the observed $\Delta k_{22}\approx-4\%$ at the
tidal frequency of Io, which is also in line with previous studies (Idini &
Stevenson, 2021; Lai, 2021). The tidal dissipation is significantly enhanced
by the presence of a compact rigid core model or an extended dilute core with
respect to the full polytrope, leading to comparable tidal quality factor $Q$
as observed (Lainey et al., 2009).

For the tidal responses to the $\Psi_{4}^{2}$ component, all of models we
considered are difficult to give rise to $\Delta k_{42}\approx-11\%$ near the
tidal frequency of Io. For the interior model with a compact rigid core,
significant dynamical corrections are generated at the tidal frequency around
$\omega/\Omega\approx-1.1$ due to the resonance with an inertial mode whose
gravitational perturbations are dominated by the spherical harmonics of $l=4$
and $m=2$. However, this resonance is too far away from the tidal frequencies
of Galilean moons. For the interior model with an extended dilute core, we
demonstrate that the gravity modes in the dilute core can be significantly
modified by the Coriolis force, leading to the mixed gravito-inertial modes.
Resonances with gravito-inertial modes in the dilute core can produce non-
negligible dynamical corrections, but they are insufficient to explain the
observed $\Delta k_{42}\approx-11\%$ near the tidal frequency of Io based on
our simplified model. We also briefly investigated the effect of a top stable
layer on Jupiter’s tides. The thin stable layer acts as a ”barrier” and tends
to restrict the density and velocity perturbations mainly in the outer
envelope. However, our numerical results show that the top stable layer has
little influence on the real part of tidal Love numbers.

As we have mentioned, we do not aim to construct a realistic interior model of
Jupiter in this study. These simplified models are designed to characterize
the tidal responses of some possible scenarios of Jupiter’s interior. Because
the dynamical tides highly depend on the tidal frequency, the satellite
dependent tidal Love numbers would provide more constraints on the interior of
Jupiter (Idini & Stevenson, 2022b). In addition, seismology is the most
effective approach to determine the interior structure of planets, though the
detection of Jupiter’s oscillations remains a big challenge (Gaulme et al.,
2011). Nevertheless, the numerical scheme we developed in this study can be
also used for theoretical calculations of oscillation modes of giant planets.

There are some caveats, which should be considered in future. First, we do not
consider the centrifugal deformation in order to solve the problem in the
spherical geometry. The centrifugal effect plays a significant role in the
tidal Love numbers of Jupiter, especially for the high-degree tidal
components. Although we have made the centrifugal corrections when the
numerical results are qualitatively compared with the observations, both the
Coriolis and centrifugal effects should be self-consistently taken into
account for quantitative comparisons with the high precision observations in
future. Second, giant planets exhibit differential rotations, which also
influence the oscillation modes and thus tidal responses (Dewberry et al.,
2021). Finally, Jupiter has the strongest magnetic field among planets in the
solar system and mainly consists of electrically conducting fluid (metallic
hydrogen), so the magnetic effects (Lin & Ogilvie, 2018; Wei, 2022) should
also play a part in the tides of Jupiter.

###### Acknowledgements.

The author would like to thank an anonymous referee for constructive comments
and Dali Kong for fruitful discussions. This study was supported by the B-type
Strategic Priority Program of the CAS (XDB41000000), National Natural Science
Foundation of China (grant no. 42174215) and the pre-research project on Civil
Aerospace Technologies of CNSA (D020308). Numerical calculations were
performed on the Taiyi cluster supported by the Center for Computational
Science and Engineering of Southern University of Science and Technology.

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