## Periodic Reporting for period 3 - RotaNut (Rotation and Nutation of a wobbly Earth)

Reporting period: 2018-09-01 to 2020-02-29

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.

In practice, the adopted model has been simplified. 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.

In practice, the adopted model has been simplified. 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

- Non-linear interaction of inertial modes

We have studied numerically the resonances of inertial modes (induced by Coriolis forces) in a spherical shell as the Earth's core, containing an inner core differentially rotating. We show the existence of widespread turbulence thanks to a cascade of triad-wave interactions.

- Instabilities, turbulence and dynamo generated by precession of a planet with an inner core.

We have considered a large number of magneto-hydrodynamic simulations of precessing spherical shells, where all the parameters have been systematically varied to study the forced basic flows, the associated instabilities, and the dynamo capability of these flows.

- Understanding the core flow when considering Coriolis, viscous, and magnetic forces with KORE

We have developed a code to solve numerically the Navier-Stokes equation for fluids in rapidly rotating, near-spheroidal cavities, forced by a wobbling mantle. We have implemented the magnetic part which allows them to study the Ohmic dissipation in the fluid. We have performed the first high-resolution numerical model that solves simultaneously the rigid body dynamics of the mantle and the Navier-Stokes equation for the liquid core. This method takes naturally into account dissipative processes in the fluid that are ignored in current nutation models.

- Geostrophic flow induced by harmonic forcing of a rotating cavity

Harmonic forcing on a rotating fluid can drive intense axisymmetric flow. A general weakly non-linear analytical theory for any multipolar forcing on a rotating sphere has been derived in the limit where both the amplitude of the forcing and the Ekman number are small. The analytical result shows that this drives an axisymmetric flow in the bulk due to the nonlinear advection in the Ekman layer at the outer boundary.

- Application to Enceladus sub-surface ocean

Various observations from Cassini have pointed out the existence of a very active global liquid subsurface water ocean in Enceladus and of geysers related to heat flux. We studied viscous dissipation in the ocean when considering libration. We showed that this still insufficient to explain the observed heat flux.

- Core-mantle boundary topography effects on nutations

In the theory of nutation of the non-rigid Earth, there is a series of relatively small effects that have to be taken. Using an analytical approach, we have added the topographic coupling focusing on resonance effects induced by the inertial modes.

- Data analysis

For interpreting variations in the magnitude and direction of the Earth’s rotation vector, some Basic Earth Parameters (BEP) from recent VLBI (Very Long Baseline Interferometry) series, namely the coupling constants at the liquid core interfaces, were computed. An updated nutation series has been built producing new forced nutation amplitudes, together with varying Free Core Nutation (FCN) amplitudes.

We have studied numerically the resonances of inertial modes (induced by Coriolis forces) in a spherical shell as the Earth's core, containing an inner core differentially rotating. We show the existence of widespread turbulence thanks to a cascade of triad-wave interactions.

- Instabilities, turbulence and dynamo generated by precession of a planet with an inner core.

We have considered a large number of magneto-hydrodynamic simulations of precessing spherical shells, where all the parameters have been systematically varied to study the forced basic flows, the associated instabilities, and the dynamo capability of these flows.

- Understanding the core flow when considering Coriolis, viscous, and magnetic forces with KORE

We have developed a code to solve numerically the Navier-Stokes equation for fluids in rapidly rotating, near-spheroidal cavities, forced by a wobbling mantle. We have implemented the magnetic part which allows them to study the Ohmic dissipation in the fluid. We have performed the first high-resolution numerical model that solves simultaneously the rigid body dynamics of the mantle and the Navier-Stokes equation for the liquid core. This method takes naturally into account dissipative processes in the fluid that are ignored in current nutation models.

- Geostrophic flow induced by harmonic forcing of a rotating cavity

Harmonic forcing on a rotating fluid can drive intense axisymmetric flow. A general weakly non-linear analytical theory for any multipolar forcing on a rotating sphere has been derived in the limit where both the amplitude of the forcing and the Ekman number are small. The analytical result shows that this drives an axisymmetric flow in the bulk due to the nonlinear advection in the Ekman layer at the outer boundary.

- Application to Enceladus sub-surface ocean

Various observations from Cassini have pointed out the existence of a very active global liquid subsurface water ocean in Enceladus and of geysers related to heat flux. We studied viscous dissipation in the ocean when considering libration. We showed that this still insufficient to explain the observed heat flux.

- Core-mantle boundary topography effects on nutations

In the theory of nutation of the non-rigid Earth, there is a series of relatively small effects that have to be taken. Using an analytical approach, we have added the topographic coupling focusing on resonance effects induced by the inertial modes.

- Data analysis

For interpreting variations in the magnitude and direction of the Earth’s rotation vector, some Basic Earth Parameters (BEP) from recent VLBI (Very Long Baseline Interferometry) series, namely the coupling constants at the liquid core interfaces, were computed. An updated nutation series has been built producing new forced nutation amplitudes, together with varying Free Core Nutation (FCN) amplitudes.

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.

Furthermore, the positioning method (based on measurements for which precise Earth rotation is absolutely necessary) is needed in order to identify where the surface of the Earth is responding to extreme conditions such as those regions susceptible to flooding and droughts, earthquakes, etc. Our global society wants to monitor these changes in the Earth system. Further, governments require this kind of information to plan as well as counteract accordingly on a local, regional, national, and international level.

We expect that our work will significantly contribute to and improve the determination of the Earth rotation, and thus satisfy the GGOS (Global Geodetic Observing System) requirements, as stated here above, in particular in the domain of dynamic Earth processes. The liquid core of the Earth is our own central target.

In addition, it is worth mentioning that our development will further help understanding the deep interior of the other terrestrial planet like Mars. We are indeed developing an instrument to measure the nutation of Mars and therewith better understand Mars interior.

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.

Furthermore, the positioning method (based on measurements for which precise Earth rotation is absolutely necessary) is needed in order to identify where the surface of the Earth is responding to extreme conditions such as those regions susceptible to flooding and droughts, earthquakes, etc. Our global society wants to monitor these changes in the Earth system. Further, governments require this kind of information to plan as well as counteract accordingly on a local, regional, national, and international level.

We expect that our work will significantly contribute to and improve the determination of the Earth rotation, and thus satisfy the GGOS (Global Geodetic Observing System) requirements, as stated here above, in particular in the domain of dynamic Earth processes. The liquid core of the Earth is our own central target.

In addition, it is worth mentioning that our development will further help understanding the deep interior of the other terrestrial planet like Mars. We are indeed developing an instrument to measure the nutation of Mars and therewith better understand Mars interior.