arXiv subjects

Adrian J. Barker

Publications and source records attributed to Adrian J. Barker.

At least 19 recordsLinked to original sources

arXiv2026

Tidal dissipation in magnetised, rotating stars and planets: linear calculations exploring various magnetic field configurations

We study tidal flows in the convective envelopes of rotating, magnetised fluid bodies, such as low-mass stars and giant planets. In well-mixed convective regions, (magneto-)inertial waves are linearly excited by tidal forcing, and their dissipation can dominantly drive spin and orbital evolution in many close star-planet and binary star systems. We perform linear magnetohydrodynamic calculations of wavelike tides in spherical-shell geometry of a tidally-forced, rotating, incompressible, viscous and non-ideal magnetised fluid. Our calculations consider the widest range of magnetic field configurations to date (including both aligned and misaligned dipole fields, free-decay dipole and quadrupole fields, azimuthal "Malkus fields" and mixed poloidal-toroidal "Prendergast fields") to analyse the effects of magnetic fields on the wavelike response and dissipation. We find that the tidal response at a given frequency depends strongly on both magnetic field strength and geometry. Magnetic fields with strong poloidal components modify the flow more efficiently and introduce high-frequency Alfvénic resonances associated with weakly damped eigenmodes. When an enhanced (turbulent) viscosity is adopted, we find that viscous dissipation remains comparable to Ohmic dissipation for strong fields, in contrast to previous studies in which Ohmic dissipation was argued to dominate. We also explore the variation in magnetic effects as the shell thickness, magnetic Prandtl and Ekman numbers are varied. Finally, the frequency-averaged tidal power is found to be largely insensitive to the magnetic field in most cases, though significant deviations are found for free-decay fields. Our results have important implications for the tidal evolution of magnetised, rotating stars and planets.

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arXiv2026

Observational imprints of tidal internal gravity wave dissipation in star-planet systems

Tidal interactions play a crucial role in the orbital evolution of close-in star-planet systems. There are numerous manifestations of tides, including planetary orbital migration, breaking resonant chains, tidal heating, orbital circularization, spin-orbit alignment, and stellar and planetary spin synchronization. In the present study, we focus on the dissipation of internal gravity waves within stars. We examine two mechanisms: wave breaking in stars with radiative cores and magnetic wave conversion in stars with convective cores. Applying tidal prescriptions modelling these processes, we demonstrate that the enhanced stellar rotation of both TOI-2458 and GJ 504 can be explained by the previous engulfment of a hot Jupiter caused by gravity wave damping. Furthermore, we show that the observed population of hot Jupiters can be divided into two distinct subsamples: those that are too young for gravity wave dissipation and those where it is ongoing. These subsamples exhibit qualitatively different orbital period distributions: young systems have a uniform distribution, while older systems show a steep decline at short orbital periods. Using a population synthesis approach, we successfully reproduce the main features of the older hot Jupiter sample based on the distribution of the younger systems. According to our estimates, up to 13% of the main-sequence stars within the mass range [0.7,1.5] $M_{\odot}$ that once hosted a hot Jupiter may have since engulfed it. Our results highlight the key role of internal gravity wave dissipation in shaping the orbital architectures of hot Jupiter systems.

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arXiv2026

The stability of propagating plane inertial waves in rotating fluids

Inertial waves transport energy and momentum in rotating fluids and are a major contributor to mixing and tidal dissipation in Earth's oceans, gaseous planets, and stellar interiors. However, their stability and breakdown mechanisms are not fully understood. We examine the linear stability and nonlinear breakdown of finite-amplitude propagating plane inertial waves using Floquet theory and direct numerical simulations. The Floquet analysis generalizes previous studies as it is valid for arbitrary perturbation wavelengths and primary wave amplitudes. We find that the wavenumber orientation of the most unstable perturbations depends strongly on the wave frequency and weakly on the wave amplitude. The most unstable perturbations have wavelengths that are small relative to the primary wave wavelength for low wave amplitudes, but become comparable for large wave amplitudes. We then use direct numerical simulations to follow the nonlinear breakdown of the wave and examine how the wave energy is either dissipated in a forward cascade or accumulated into long-lived geostrophic modes. Simulations reveal that the conversion efficiency into geostrophic modes increases with increasing wave amplitude, as expected for pumping of geostrophic modes by nearly-resonant triadic interactions. We also find that the conversion efficiency increases with decreasing primary wave frequency, which may be due to the more efficient coupling of quasi-2D waves to geostrophic modes. These results on the stability and breakdown of single plane inertial waves provides additional foundation for understanding the role of inertial waves in rotating turbulence, transport properties of inertial wave beams, and inertial wave propagation in more complex environments such as those with magnetic fields or shear flows.

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arXiv2026

Chasing the Tides: Searching for Orbital Decay Signatures in Transit Timing Data and Tidal Models for 20 Hot Jupiters

In this work, we present a transit timing variation analysis for 20 hot Jupiter systems, which we interpret with theoretical tidal dissipation models. For the majority of the sample, we conclude that a constant orbital period model represents the timing data best. Only WASP-12 b, TrES-1 b and WASP-121 b exhibit a changing orbital period, according to the most up-to-date results. We updated the orbital decay rate of WASP-12 b to $\dot{P} = -29.4 \pm 4.0 \mathrm{~ms~yr^{-1}}$ and the corresponding stellar tidal quality factor to $Q_*^{\prime} = 1.72 \pm 0.18 \times 10^5$. For TrES-1 b, the median quadratic model suggests a period decrease at a rate of $-14.9 \pm 0.6 \mathrm{~ms~yr^{-1}}$, but the corresponding $Q_*^{\prime} = 570 \pm 60$ does not agree with the theoretical estimates, which suggest $Q_*^{\prime} \sim 10^6$ due to internal gravity wave dissipation. Lastly, WASP-121 b exhibits orbital growth at a rate of $15.1 \pm 0.8 \mathrm{~ms~yr^{-1}}$, and theoretical results support outward migration due to strong inertial wave dissipation.

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arXiv2025

Tidal dissipation and spin-orbit alignment due to the precessional instability in convection zones in rotating giant planets and stars

Tidal dissipation in star-planet systems occurs through various mechanisms, including the precessional instability. This is an instability of laminar flows (``Poincaré flows") forced by axial precession of a rotating, oblate, spin-orbit misaligned fluid planet or star, which excites inertial waves in convective regions if the dimensionless precession rate (``Poincaré number" $\mathrm{Po}$) is sufficiently large. We constrain the contribution of the precessional instability to tidal dissipation and heat transport, using Cartesian hydrodynamical simulations in a small patch of a planet, and study its interaction with turbulent convection, modelled as rotating Rayleigh-Bénard convection. The precessional instability without convection results in laminar flow at low values and turbulent flow at sufficiently high values of $\mathrm{Po}$. The associated tidal dissipation rate scales as $\mathrm{Po}^2$ and $\mathrm{Po}^3$ in each regime, respectively. With convection, the Poincaré number at which turbulent flow is achieved shifts to lower values for stronger convective driving. Convective motions also act on large-scale tidal flows like an effective viscosity, resulting in continuous tidal dissipation (scaling as $\mathrm{Po}^2$), which obfuscates or suppresses tidal dissipation due to precessional instability. The effective viscosities obtained agree with scaling laws previously derived using (rotating) mixing-length theory. By evaluating our scaling laws using interior models of Hot Jupiters, we find that the precessional instability is significantly more efficient than the effective viscosity of convection. The former drives alignment in 1 Gyr for a Jupiter-like planet orbiting within 23 days. Linearly excited inertial waves can be even more effective for wider orbits, aligning spins for orbits within 53-142 days.

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arXiv2025

Interplay between tidal flows and magnetic fields in nonlinear simulations of stellar and planetary convective envelopes

Stars and planets in close systems are magnetised but the influence of magnetic fields on their tidal responses (and vice versa) and dissipation rates has not been well explored. We present exploratory nonlinear magnetohydrodynamical (MHD) simulations of tidally-excited inertial waves in convective envelopes. These waves probably provide the dominant contribution to tidal dissipation in several astrophysical settings, including tidal circularisation of solar-type binary stars and hot Jupiters, and orbital migration of the moons of Jupiter and Saturn. We model convective envelopes as incompressible magnetised fluids in spherical shells harbouring an initially (rotationally-aligned) dipolar magnetic field. We find that depending on its strength (quantified by its Lehnert number Le) and the magnetic Prandtl number Pm, the magnetic field can either deeply modify the tidal response or be substantially altered by tidal flows. Simulations with small Le exhibit strong tidally-generated differential rotation (zonal flows) for sufficiently large tidal amplitudes, such that both the amplitude and topology of the initial magnetic field are tidally impacted. In contrast, strong magnetic fields can inhibit these zonal flows through large-scale magnetic torques, and by Maxwell stresses arising from magneto-rotational instability, which we identify and characterise in our simulations, along with the role of torsional Alfvén waves. Without tidally-driven zonal flows, the resulting tidal dissipation is close to the linear predictions. We quantify the transition Le as a function of Pm, finding it to be comparable to realistic values found in solar-like stars, such that we predict complex interactions between tidal flows and magnetic fields.

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arXiv2025

Tidal interactions in stellar and planetary systems

Gravitational tidal interactions drive long-term rotational and orbital evolution in planetary systems, in multiple (particularly close binary) star systems and in planetary moon systems. Dissipation of tidal flows in Earth's oceans is primarily responsible for producing gradual expansion of the Moon's orbit at a few centimetres per year as the Earth's day lengthens by a few milliseconds per century. Similar processes occur in many astrophysical systems. For example, tidal dissipation inside (slowly rotating) stars hosting short-period planets can cause the orbits of these planets to decay, potentially leading to planetary destruction; tidal dissipation inside stars in close stellar binary systems -- and inside short-period planets such as hot Jupiters in planetary systems -- can cause initially eccentric orbits to become circular. To model these processes, explain many current observational results, and make predictions for future observations, we require a detailed theoretical understanding of tidal flows and the mechanisms by which -- and how efficiently -- they are dissipated inside stars and planets. This article will introduce our current understanding of tidal flows and dissipation inside stars (and to a lesser extent giant planets), as well as highlight some unsolved problems.

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arXiv2024

Interactions between tidal flows and magnetic fields in stellar/planetary convective envelopes

Stars and gaseous planets are magnetised objects but the influence of magnetic fields on their tidal responses and dissipation rates has not been well explored. We present the first exploratory nonlinear magnetohydrodynamic (MHD) simulations of tidally-excited waves in incompressible convective envelopes harbouring an initial dipolar magnetic field. Simulations with weak magnetic fields exhibit tidally-generated differential rotation in the form of zonal flows (like in the purely hydrodynamic case) that can modify tidal dissipation rates from prior linear predictions. Moreover, tidal waves and zonal flows affect the amplitude and structure of the magnetic field, notably through creation of toroidal fields via the $Ω$-effect. In contrast, simulations with strong magnetic fields feature severely inhibited zonal flows, due to large-scale magnetic stresses, excitation of torsional waves, or magnetic instabilities. We predict that the different regimes observed for weak and strong magnetic fields may be both relevant for low-mass stars when using turbulent values of the magnetic Prandtl number.

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arXiv2024

Local stability of differential rotation in magnetised radiation zones and the solar tachocline

We study local magnetohydrodynamical (MHD) instabilities of differential rotation in magnetised, stably-stratified regions of stars and planets using a Cartesian Boussinesq model. We consider arbitrary latitudes and general shears (with gravity direction misaligned from this by an angle $ϕ$), to model radial ($ϕ=0$), latitudinal ($ϕ=\pm 90^\circ$), and mixed differential rotations, and study both non-diffusive (including magnetorotational, MRI, and Solberg-Høiland instabilities) and diffusive instabilities (including Goldreich-Schubert-Fricke, GSF, and MRI with diffusion). These instabilities could drive turbulent transport and mixing in radiative regions, including the solar tachocline and the cores of red giant stars, but their dynamics are incompletely understood. We revisit linear axisymmetric instabilities with and without diffusion and analyse their properties in the presence of magnetic fields, including deriving stability criteria and computing growth rates, wavevectors and energetics, both analytically and numerically. We present a more comprehensive analysis of axisymmetric local instabilities than prior work, exploring arbitrary differential rotations and diffusive processes. The presence of a magnetic field leads to stability criteria depending upon angular velocity rather than angular momentum gradients. We find MRI operates for much weaker differential rotations than the hydrodynamic GSF instability, and that it typically prefers much larger lengthscales, while the GSF instability is impeded by realistic strength magnetic fields. We anticipate MRI to be more important for turbulent transport in the solar tachocline than the GSF instability when $ϕ>0$ in the northern (and vice versa in the southern) hemisphere, though the latter could operate just below the convection zone when MRI is absent for $ϕ<0$.

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arXiv2024

An efficient tidal dissipation mechanism via stellar magnetic fields

Recent work suggests that inwardly propagating internal gravity waves (IGWs) within a star can be fully converted to outward magnetic waves (MWs) if they encounter a sufficiently strong magnetic field. The resulting magnetic waves dissipate as they propagate outward to regions with lower Alfvén velocity. While tidal forcing is known to excite IGWs, this conversion and subsequent damping of magnetic waves has not been explored as a tidal dissipation mechanism. In particular, stars with sufficiently strong magnetic fields could fully dissipate tidally excited waves, yielding the same tidal evolution as the previously-studied ``travelling wave regime''. Here, we evaluate the viability of this mechanism using stellar models of stars with convective cores (F-type stars in the mass range of $1.2$-$1.6M_\odot$) which were previously thought to be weakly tidally dissipative (due to the absence of nonlinear gravity wave breaking). The criterion for wave conversion to operate is evaluated for each stellar mass using the properties of each star's interior along with estimates of the magnetic field produced by a convective core dynamo under the assumption of equipartition between kinetic (convective) and magnetic energies. Our main result is that this previously unexplored source of efficient tidal dissipation can operate in stars within this mass range for significant fractions of their lifetimes. This tidal dissipation mechanism appears to be consistent with the observed inspiral of WASP-12b, and more generally could play an important role in the orbital evolution of hot Jupiters -- and to lower mass ultra-short period planets -- orbiting F-type stars.

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arXiv2023

Tidal dissipation in rotating and evolving giant planets with application to exoplanet systems

We study tidal dissipation in models of rotating giant planets with masses in the range $0.1 - 10 M_\mathrm{J}$ throughout their evolution. Our models incorporate a frequency-dependent turbulent effective viscosity acting on equilibrium tides (including its modification by rapid rotation consistent with hydrodynamical simulations) and inertial waves in convection zones, and internal gravity waves in the thin radiative atmospheres. We consider a range of planetary evolutionary models for various masses and strengths of stellar instellation. Dissipation of inertial waves is computed using a frequency-averaged formalism fully accounting for planetary structures. Dissipation of gravity waves in the radiation zone is computed assuming these waves are launched adiabatically and are subsequently fully damped (by wave breaking/radiative damping). We compute modified tidal quality factors $Q'$ and evolutionary timescales for these planets as a function of their ages. We find inertial waves to be the dominant mechanism of tidal dissipation in giant planets whenever they are excited. Their excitation requires the tidal period ($P_\mathrm{tide}$) to be longer than half the planetary rotation ($P_\mathrm{rot}/2$), and we predict inertial waves to provide a typical $Q'\sim 10^3 (P_\mathrm{rot}/1 \mathrm{d})^2$, with values between $10^5$ and $10^6$ for a 10-day period. We show correlations of observed exoplanet eccentricities with tidal circularisation timescale predictions, highlighting the key role of planetary tides. A major uncertainty in planetary models is the role of stably-stratified layers resulting from compositional gradients, which we do not account for here, but which could modify predictions for tidal dissipation rates.

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arXiv2023

On the orbital decay of the gas giant Kepler-1658b

The gas giant Kepler-1658b has been inferred to be spiralling into its sub-giant F-type host star Kepler-1658a (KOI-4). The measured rate of change of its orbital period is $\dot{P}_{\rm orb}=-131^{+20}_{-22}\mathrm{ms/yr}$, which can be explained by tidal dissipation in the star if its modified tidal quality factor is as low as $Q^{\,\prime}\approx 2.50\times {10}^{4}$. We explore whether this could plausibly be consistent with theoretical predictions based on applying up-to-date tidal theory in stellar models (varying stellar mass, age, and metallicity) consistent with our newly-derived observational constraints. In most of our models matching the combined constraints on the stellar effective temperature and radius, the dissipation in the star is far too weak, capable of providing $Q^{\,\prime}\gtrsim 10^9$, hence contributing negligibly to orbital evolution. Using only constraints on the stellar radius, efficient tidal dissipation sufficient to explain observations is possible due to inertial waves in the convective envelope during the sub-giant phase, providing $Q^{\,\prime}\sim 10^4$, but this period in the evolution is very short-lived (shorter than $10^2$ yrs in our models). We show that dissipation in the planet is capable of explaining the observed $\dot{P}_\mathrm{orb}$ only if the planet rotates non-synchronously. Tidally-induced pericentre precession is a viable explanation if the periastron argument is near $3π/2$ and the quadrupolar Love number is above 0.26. Further observations constraining the stellar and planetary properties in this system have the exciting potential to test tidal theories in stars and planets.

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arXiv2023

Tidal dissipation in stably stratified and semi-convective regions of rotating giant planets: incorporating Coriolis forces

We study how stably stratified or semi-convective layers alter tidal dissipation rates associated with the generation of inertial, gravito-inertial, interfacial and surface gravity waves in rotating giant planets. We explore scenarios in which stable (non-convective) layers contribute to the high rates of tidal dissipation observed for Jupiter and Saturn in our solar system. Our model is an idealised spherical Boussinesq system incorporating Coriolis forces to study effects of stable stratification and semi-convective layers on tidal dissipation. Our detailed numerical calculations consider realistic tidal forcing and compute the resulting viscous and thermal dissipation rates. The presence of an extended stably stratified fluid core significantly enhances tidal wave excitation of both inertial waves (due to rotation) in the convective envelope and gravito-inertial waves in the dilute core. We show that a sufficiently strongly stratified fluid core enhances inertial wave dissipation in a convective envelope much like a solid core does. We demonstrate that efficient tidal dissipation rates (and associated tidal quality factors $Q'$) -- sufficient to explain the observed migration rates of Saturn's moons -- are predicted at the frequencies of the orbiting moons due to the excitation of inertial or gravito-inertial waves in our models with stable layers (without requiring resonance-locking). Stable layers could also be important for tidal evolution of hot and warm Jupiters, and hot Neptunes, providing efficient tidal circularisation rates. Future work should study more sophisticated planetary models that also account for magnetism and differential rotation, as well as the interaction of inertial waves with turbulent convection.

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arXiv2023

Do nonlinear effects disrupt tidal dissipation predictions in convective envelopes?

Most prior works studying tidal interactions in tight star/planet or star/star binary systems have employed linear theory of a viscous fluid in a uniformly-rotating two-dimensional spherical shell. However, compact systems may have sufficiently large tidal amplitudes for nonlinear effects to be important. We compute tidal flows subject to nonlinear effects in a 3D, thin (solar-like) convective shell, spanning the entire frequency range of inertial waves. Tidal frequency-averaged dissipation predictions of linear theory with solid body rotation are approximately reproduced in our nonlinear simulations (though we find it to be reduced by a factor of a few), but we find significant differences, potentially by orders of magnitude, at a fixed tidal frequency corresponding to a specific two-body system at a given epoch. This is largely due to tidal generation of differential rotation (zonal flows) and their effects on the waves.

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arXiv2023

Tidally-excited inertial waves in stars and planets: exploring the frequency-dependent and averaged dissipation with nonlinear simulations

We simulate the nonlinear hydrodynamical evolution of tidally-excited inertial waves in convective envelopes of rotating stars and giant planets modelled as spherical shells containing incompressible, viscous and adiabatically-stratified fluid. This model is relevant for studying tidal interactions between close-in planets and their stars, as well as close low-mass star binaries. We explore in detail the frequency-dependent tidal dissipation rates obtained from an extensive suite of numerical simulations, which we compare with linear theory, including with the widely-employed frequency-averaged formalism to represent inertial wave dissipation. We demonstrate that the frequency-averaged predictions appear to be quite robust and is approximately reproduced in our nonlinear simulations spanning the frequency range of inertial waves as we vary the convective envelope thickness, tidal amplitude, and Ekman number. Yet, we find nonlinear simulations can produce significant differences with linear theory for a given tidal frequency (potentially by orders of magnitude), largely due to tidal generation of differential rotation and its effects on the waves. Since the dissipation in a given system can be very different both in linear and nonlinear simulations, the frequency-averaged formalism should be used with caution. Despite its robustness, it is also unclear how accurately it represents tidal evolution in real (frequency-dependent) systems.

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arXiv2023

Tidal dissipation due to the elliptical instability and turbulent viscosity in convection zones in rotating giant planets and stars

Tidal dissipation in star-planet systems can occur through various mechanisms, among which is the elliptical instability. This acts on elliptically deformed equilibrium tidal flows in rotating fluid planets and stars, and excites inertial waves in convective regions if the dimensionless tidal amplitude ($ε$) is sufficiently large. We study its interaction with turbulent convection, and attempt to constrain the contributions of both elliptical instability and convection to tidal dissipation. For this, we perform an extensive suite of Cartesian hydrodynamical simulations of rotating Rayleigh-Bénard convection in a small patch of a planet. We find that tidal dissipation resulting from the elliptical instability, when it operates, is consistent with $ε^3$, as in prior simulations without convection. Convective motions also act as an effective viscosity on large-scale tidal flows, resulting in continuous tidal dissipation (scaling as $ε^2$). We derive scaling laws for the effective viscosity using (rotating) mixing-length theory, and find that they predict the turbulent quantities found in our simulations very well. In addition, we examine the reduction of the effective viscosity for fast tides, which we observe to scale with tidal frequency ($ω$) as $ω^{-2}$. We evaluate our scaling laws using interior models of Hot Jupiters computed with MESA. We conclude that rotation reduces convective length scales, velocities and effective viscosities (though not in the fast tides regime). We estimate that elliptical instability is efficient for the shortest-period Hot Jupiters, and that effective viscosity of turbulent convection is negligible in giant planets compared with inertial waves.

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arXiv2023

Linear and nonlinear properties of the Goldreich-Schubert-Fricke instability in stellar interiors with arbitrary local radial and latitudinal differential rotation

We investigate the linear and nonlinear properties of the Goldreich-Schubert-Fricke (GSF) instability in stellar radiative zones with arbitrary local (radial and latitudinal) differential rotation. This instability may lead to turbulence that contributes to redistribution of angular momentum and chemical composition in stars. In our local Boussinesq model, we investigate varying the orientation of the shear with respect to the 'effective gravity', which we describe using the angle $ϕ$. We first perform an axisymmetric linear analysis to explore the effects of varying $ϕ$ on the local stability of arbitrary differential rotations. We then explore the nonlinear hydrodynamical evolution in three dimensions using a modified shearing box. The model exhibits both the diffusive GSF instability, and a non-diffusive instability that occurs when the Solberg-H\{o}iland criteria are violated. We observe the nonlinear development of strong zonal jets ("layering" in the angular momentum) with a preferred orientation in both cases, which can considerably enhance turbulent transport. By varying $ϕ$ we find the instability with mixed radial and latitudinal shears transports angular momentum more efficiently (particularly if adiabatically unstable) than cases with purely radial shear $(ϕ= 0)$. By exploring the dependence on box size, we find the transport properties of the GSF instability to be largely insensitive to this, implying we can meaningfully extrapolate our results to stars. However, there is no preferred length-scale for adiabatic instability, which therefore exhibits strong box-size dependence. These instabilities may contribute to the missing angular momentum transport required in red giant and subgiant stars and drive turbulence in the solar tachocline.

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arXiv2023

Tidal dissipation in stratified and semi-convective regions of giant planets

We study how stably stratified or semi-convective layers alter the tidal dissipation rates associated with the generation of internal waves in planetary interiors. We consider if these layers could contribute to the high rates of tidal dissipation observed for Jupiter and Saturn in our solar system. We use an idealised global spherical Boussinesq model to study the influence of stable stratification and semi-convective layers on tidal dissipation rates. We carry out analytical and numerical calculations considering realistic tidal forcing and measure how the viscous and thermal dissipation rates depend on the parameters relating to the internal stratification profile. We find that the strongly frequency-dependent tidal dissipation rate is highly dependent on the parameters relating to the stable stratification, with strong resonant peaks that align with the internal modes of the system. The locations and sizes of these resonances depend on the form and parameters of the stratification, which we explore both analytically and numerically. Our results suggest that stable stratification can significantly enhance the tidal dissipation in particular frequency ranges. Analytical calculations in the low frequency regime give us scaling laws for the key parameters, including the tidal quality factor $Q'$ due to internal gravity waves. Stably stratified layers can significantly contribute to tidal dissipation in solar and extrasolar giant planets, and we estimate substantial tidal evolution for hot Neptunes. Further investigation is needed to robustly quantify the significance of the contribution in realistic interior models, and to consider the contribution of inertial waves.

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