## Periodic Reporting for period 4 - RotaNut (Rotation and Nutation of a wobbly Earth)

Reporting period: 2020-03-01 to 2020-08-31

The relationship between the celestial frame and the terrestrial frames is complicated by the fact that the rotation and orientation of the Earth is subject to irregularities. The research proposed will result in the development of improved model of Earth rotation and orientation with an unprecedented accuracy – at the sub-centimetre level.

The Earth orientation changes are caused by the gravitational pull of the Sun and the Moon, as well as by many other factors that are progressively being identified by geodesists and geophysicists (in particular, the existence of a liquid core inside the Earth plays an important role). Because the Earth’s shape can approximately be described as an ellipsoid flattened at its poles, the combined forces acting upon the Earth produce changes in both the speed of rotation and the orientation of the axis of spin. The term ‘precession’ describes the long-term trend of this latter motion, while ‘nutation’ is the name given to shorter-term periodic variations, which are the prime focus of the present project. The precession of the Earth in space corresponds to about 50 arcseconds (1 arcsecond =1”=1°/3600) per year and the nutation amplitude is at the level of a few tens of arcseconds. The rotation axis of the Earth is moving in space at the level of 1.5km/year due to precession and has periodic variations at the level of 600 metres (as seen from space in a plane tangent to the pole). The present observations allow scientists to measure these at the centimetre level.

Earth rotation changes, precession and nutation are measured using Very Long Baseline Interferometry (VLBI), a technique that employs huge radio telescopes to observe extra-galactic radio sources such as quasars and providing the realisation of the celestial reference frame. There are presently significant differences at a few centimetres level between the VLBI observations and the results obtained from applying a theoretical model adopted by the International Astronomical Union (IAU) in 2000 and by the International Union of Geodesy and Geophysics (IUGG) in 2003.

The aim of our project RotaNut is to improve the Earth rotation modelling and to get further insight into the Earth’s interior. By now, we have understood that there are important contributions from the dynamics and the dissipation in the liquid core and at the boundaries and the presence of a topography at the core-mantle boundary at the kilometre level play an important role as well.

What is the problem/issue being addressed? & Why is it important for society? See https://rotanut.oma.be/

What are the overall objectives? See https://rotanut.oma.be/objectives

The Earth orientation changes are caused by the gravitational pull of the Sun and the Moon, as well as by many other factors that are progressively being identified by geodesists and geophysicists (in particular, the existence of a liquid core inside the Earth plays an important role). Because the Earth’s shape can approximately be described as an ellipsoid flattened at its poles, the combined forces acting upon the Earth produce changes in both the speed of rotation and the orientation of the axis of spin. The term ‘precession’ describes the long-term trend of this latter motion, while ‘nutation’ is the name given to shorter-term periodic variations, which are the prime focus of the present project. The precession of the Earth in space corresponds to about 50 arcseconds (1 arcsecond =1”=1°/3600) per year and the nutation amplitude is at the level of a few tens of arcseconds. The rotation axis of the Earth is moving in space at the level of 1.5km/year due to precession and has periodic variations at the level of 600 metres (as seen from space in a plane tangent to the pole). The present observations allow scientists to measure these at the centimetre level.

Earth rotation changes, precession and nutation are measured using Very Long Baseline Interferometry (VLBI), a technique that employs huge radio telescopes to observe extra-galactic radio sources such as quasars and providing the realisation of the celestial reference frame. There are presently significant differences at a few centimetres level between the VLBI observations and the results obtained from applying a theoretical model adopted by the International Astronomical Union (IAU) in 2000 and by the International Union of Geodesy and Geophysics (IUGG) in 2003.

The aim of our project RotaNut is to improve the Earth rotation modelling and to get further insight into the Earth’s interior. By now, we have understood that there are important contributions from the dynamics and the dissipation in the liquid core and at the boundaries and the presence of a topography at the core-mantle boundary at the kilometre level play an important role as well.

What is the problem/issue being addressed? & Why is it important for society? See https://rotanut.oma.be/

What are the overall objectives? See https://rotanut.oma.be/objectives

During this ERC RotaNut Contract, we have advanced our understanding of the Earth dynamics (and of its liquid core in particular) by quantifying our planet’s rotation changes in space and time and by analysing the fluid dynamics associated with it. We have implemented the following actions:

Work on the interactions between rotational and inertial modes in the fluid core, in the case of a non-spherical core-mantle, with the help of numerical simulations

The Earth can be seen as a global oscillator resulting from the coupling between three oscillators: the (solid) inner core, the (liquid) outer core, and the (solid) mantle. Each oscillator seeks to respond to external perturbations at its preferred natural frequencies, but coupling between the layers results in the emergence of global modes where the individual modes interact. This approach necessitates coupling the Navier-Stokes equation in the core with a Liouville equation for the wobbles of the mantle. The Earth is a fast-spinning body, allowing for the possibility of inertial waves, i.e. waves that are restored by the Coriolis force. The presence of inertial waves influences dramatically the flow dynamics.

(1) We have first developed TenGSHui, a Mathematica package to handle tensor equations in slightly aspherical configurations. Tensor calculus over surfaces, required in the boundary conditions of non-hydrostatic configurations, is also implemented. Computational demands have been significantly reduced, making it possible to truncate equations at high spherical harmonic degrees. [Trinh, 2019, PhD thesis]

(2) We have developed a numerical method (code FENNEC for Framework for Easy Numeration in NEar-spherical Configurations) built on a spectral decomposition based on TenGSHui in three dimensions to compute the inertial modes of a container with near-spherical geometry. This allowed us to compute the geostrophic flow (where the pressure gradient balances the Coriolis effect) associated with harmonic forcing of a rotating cavity. We have shown that a systematic axisymmetric flow in the bulk of the fluid appears. We have demonstrated the convergence of this approach considering slight deviations from a sphere of the boundary for the triaxial ellipsoid. [Rekier et al., 2019]. We have also designed a method to derive analytical formulae for the frequencies of the free core nutation (FCN) and Chandler wobble (CW) that are valid to all orders of the dynamical flattening of the core and mantle. [Rekier et al., 2020]

(3) We have developed a numerical code (Kore) to study the core flow within rapidly rotating planets or other rotating fluids contained within near-spherical boundaries; the current version solves the linear Navier-Stokes and induction equations for a viscous, incompressible and conductive fluid with an externally imposed axial and homogeneous magnetic field, and enclosed within a coupled rotating spherical mantle. We have shown that interactions exist between the inertial modes and rotational modes. We further examined the changes in the frequencies and damping of the inertial modes and global rotation mode with the core flattening, Ekman number, and mantle moment of inertia. We have shown the existence of exchange of personality between the eigenmodes. [Triana et al., 2019] Our results have also demonstrated the roles of viscous and ohmic dissipations and where these dissipations occur in the core. [Triana et al., 2020]

Work on the role of instabilities in a precessing core, with the help of numerical simulations

(4) We have improved our understanding of instabilities generated in the liquid core by precession and nutations of the Earth and the transition between these instabilities and turbulence by using the existing numerical code Xshells. We have shown the presence of instabilities coming from the interaction between internal shear layers. We have determined the localization of the development of the instability and found that, depending of radius of the inner core, the instability will appear at different locations. Intermittent large-scale vortices have been observed in several simulations. [Cébron et al., 2018] [Houliez et al., 2020].

Work on simulations with respect to experiments

(5) The inertial wave instability observed in experiments using the same geometrical configuration as the Earth’s core (spherical-Couette), appears to be quite complex. We have shown that the inertial wave instability participates in turbulence generation mechanism in the core. [Hoff et al., 2016]. One of the prominent features of these differentially rotating spherical-Couette experiments is the high degree of non-linear coupling between eigenmodes, as described in the numerical simulations. [Barik et al., 2018].

Work on core-mantle topography effects

(6) The role of the CMB bumps/topography on the flows in the core has been examined analytically. Solving the Navier-Stokes equation for global rotational motions and inertial waves, we have shown that, for particular degrees and orders of the topography developed in spherical harmonics, inertial waves can be sufficiently close to the forcing frequencies to induce significant perturbations in the Earth orientation and rotation variations. [Puica et al., 2020] [Dehant et al., 2020]

Research starting from VLBI observations

(7) We have developed codes for nutation computation and for the determination of the basic Earth interior parameters from using VLBI observation. We have first adjusted nutation amplitudes for the largest nutations; we have then re-estimated the Basic Earth Parameters (BEP), which are related to the Earth interior, and in particular the coupling constant at the CMB. [Zhu et al., 2017] [Zhu and Dehant, 2018]

(8) Furthermore, we have re-estimated and proposed an interpretation for the free FCN mode visible in the observation. [Zhu et al., 2020a]

(9) We have then built a new nutation model for predictions, based on our new Earth interior parameters. [Zhu et al., 2020b]

Work on the interactions between rotational and inertial modes in the fluid core, in the case of a non-spherical core-mantle, with the help of numerical simulations

The Earth can be seen as a global oscillator resulting from the coupling between three oscillators: the (solid) inner core, the (liquid) outer core, and the (solid) mantle. Each oscillator seeks to respond to external perturbations at its preferred natural frequencies, but coupling between the layers results in the emergence of global modes where the individual modes interact. This approach necessitates coupling the Navier-Stokes equation in the core with a Liouville equation for the wobbles of the mantle. The Earth is a fast-spinning body, allowing for the possibility of inertial waves, i.e. waves that are restored by the Coriolis force. The presence of inertial waves influences dramatically the flow dynamics.

(1) We have first developed TenGSHui, a Mathematica package to handle tensor equations in slightly aspherical configurations. Tensor calculus over surfaces, required in the boundary conditions of non-hydrostatic configurations, is also implemented. Computational demands have been significantly reduced, making it possible to truncate equations at high spherical harmonic degrees. [Trinh, 2019, PhD thesis]

(2) We have developed a numerical method (code FENNEC for Framework for Easy Numeration in NEar-spherical Configurations) built on a spectral decomposition based on TenGSHui in three dimensions to compute the inertial modes of a container with near-spherical geometry. This allowed us to compute the geostrophic flow (where the pressure gradient balances the Coriolis effect) associated with harmonic forcing of a rotating cavity. We have shown that a systematic axisymmetric flow in the bulk of the fluid appears. We have demonstrated the convergence of this approach considering slight deviations from a sphere of the boundary for the triaxial ellipsoid. [Rekier et al., 2019]. We have also designed a method to derive analytical formulae for the frequencies of the free core nutation (FCN) and Chandler wobble (CW) that are valid to all orders of the dynamical flattening of the core and mantle. [Rekier et al., 2020]

(3) We have developed a numerical code (Kore) to study the core flow within rapidly rotating planets or other rotating fluids contained within near-spherical boundaries; the current version solves the linear Navier-Stokes and induction equations for a viscous, incompressible and conductive fluid with an externally imposed axial and homogeneous magnetic field, and enclosed within a coupled rotating spherical mantle. We have shown that interactions exist between the inertial modes and rotational modes. We further examined the changes in the frequencies and damping of the inertial modes and global rotation mode with the core flattening, Ekman number, and mantle moment of inertia. We have shown the existence of exchange of personality between the eigenmodes. [Triana et al., 2019] Our results have also demonstrated the roles of viscous and ohmic dissipations and where these dissipations occur in the core. [Triana et al., 2020]

Work on the role of instabilities in a precessing core, with the help of numerical simulations

(4) We have improved our understanding of instabilities generated in the liquid core by precession and nutations of the Earth and the transition between these instabilities and turbulence by using the existing numerical code Xshells. We have shown the presence of instabilities coming from the interaction between internal shear layers. We have determined the localization of the development of the instability and found that, depending of radius of the inner core, the instability will appear at different locations. Intermittent large-scale vortices have been observed in several simulations. [Cébron et al., 2018] [Houliez et al., 2020].

Work on simulations with respect to experiments

(5) The inertial wave instability observed in experiments using the same geometrical configuration as the Earth’s core (spherical-Couette), appears to be quite complex. We have shown that the inertial wave instability participates in turbulence generation mechanism in the core. [Hoff et al., 2016]. One of the prominent features of these differentially rotating spherical-Couette experiments is the high degree of non-linear coupling between eigenmodes, as described in the numerical simulations. [Barik et al., 2018].

Work on core-mantle topography effects

(6) The role of the CMB bumps/topography on the flows in the core has been examined analytically. Solving the Navier-Stokes equation for global rotational motions and inertial waves, we have shown that, for particular degrees and orders of the topography developed in spherical harmonics, inertial waves can be sufficiently close to the forcing frequencies to induce significant perturbations in the Earth orientation and rotation variations. [Puica et al., 2020] [Dehant et al., 2020]

Research starting from VLBI observations

(7) We have developed codes for nutation computation and for the determination of the basic Earth interior parameters from using VLBI observation. We have first adjusted nutation amplitudes for the largest nutations; we have then re-estimated the Basic Earth Parameters (BEP), which are related to the Earth interior, and in particular the coupling constant at the CMB. [Zhu et al., 2017] [Zhu and Dehant, 2018]

(8) Furthermore, we have re-estimated and proposed an interpretation for the free FCN mode visible in the observation. [Zhu et al., 2020a]

(9) We have then built a new nutation model for predictions, based on our new Earth interior parameters. [Zhu et al., 2020b]

The general objective of our science is advancing our understanding of the dynamic Earth system by quantifying our planet’s rotation changes in space and time. This is one of the most important objectives of geodesy. We live on a dynamic planet rotating in space, in constant motion requiring for its understanding long-term, continuous quantification of its changes in truly stable frames of reference, which includes understanding of Earth rotation. Our scientific objective is to better understand the Earth interior and the Earth rotation for helping the above missions. Earth rotation is a fundamental backbone for positioning, which has many other scientific and societal applications.

Our work significantly contributes to and improves the determination of the Earth rotation, and thus satisfy the GGOS (Global Geodetic Observing System) requirements in the domain of dynamic Earth processes. The liquid core of the Earth is our own central target.

In addition, we initiated the development with industry of a VLBI Transmitter that is mimicking quasar and will possibly be placed on the Galileo satellites. This will help improving the reference frames and the Earth rotation parameter determination.

Moreover, it is worth mentioning that our development will further help understanding the deep interior of the other terrestrial planet like Mars. We are indeed ready to send within the ExoMars 2022 ESA-Roscosmos mission, an instrument to measure the nutation of Mars and therewith better understand Mars interior.

Our work significantly contributes to and improves the determination of the Earth rotation, and thus satisfy the GGOS (Global Geodetic Observing System) requirements in the domain of dynamic Earth processes. The liquid core of the Earth is our own central target.

In addition, we initiated the development with industry of a VLBI Transmitter that is mimicking quasar and will possibly be placed on the Galileo satellites. This will help improving the reference frames and the Earth rotation parameter determination.

Moreover, it is worth mentioning that our development will further help understanding the deep interior of the other terrestrial planet like Mars. We are indeed ready to send within the ExoMars 2022 ESA-Roscosmos mission, an instrument to measure the nutation of Mars and therewith better understand Mars interior.