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Teaser, summary, work performed and final results

Periodic Reporting for period 2 - RotaNut (Rotation and Nutation of a wobbly Earth)

Teaser

Summary

What is the problem/issue being addressed?

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 â€“ at the sub-centimetre level.
The rotation changes are known as variations of the length of the day, which are at the level of a few tenths of millisecond (ms) per day.
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 Earthâ€™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 â€˜precessionâ€™ describes the long-term trend of this latter motion, while â€˜nutationâ€™ 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 =1â€=1Â°/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 adopted model is based on the idea that the Earth reacts as a deformable object with a deformable inner core (viscoelastic central part of the Earth composed of a solid iron alloy), a liquid core (also composed of iron alloy), a deformable viscoelastic mantle (composed mainly of olivine and perovskite), as well as oceans and an atmosphere. The adopted theoretical model is not perfect as seen from the observed residuals. Though they were obtained from a precise computation of the forcing (gravitational torques exerted by other solar system bodies on the Earth), on the one hand, and of the response of the deformable Earth to this forcing, on the other hand. In practice, the final model was obtained from a convolution between a precise rigid-Earth forcing theory and a transfer function accounting for some physical properties of the Earth interior, considering small contributions from its atmosphere. However, the Earth is a more complex object than that, and in particular, the atmospheric and oceanic contribution to Earth orientation are not perfectly modelled, and the coupling mechanisms at the boundaries between the inner core, the liquid outer core, and the mantle are not yet understood enough to be properly modelled. The aim of our project RotaNut is to improve the Earth rotation modelling and to get further insight into the Earthâ€™s interior. The study requires a multidisciplinary approach mixing fields in astronomy, geophysics, geodesy, and fluid dynamics. The research proposed shall make important strides in understanding and modelling the physical processes inside the Earth (and inside the liquid core in particular) associated with Earth rotation and nutation. The existence

Work performed

- Tensor equations in planets: get peace of mind with TenGSHui
We use a Mathematica package to handle our equations in the slightly aspherical geometries displayed by the Earth and other planets and moons. The method relies on series expansions in powers of the flattening, and we demonstrated both its formal and numerical convergence in oblate spheroids and triaxial ellipsoids, where comparison with analytical solutions is possible (see part on the inertial modes of an inviscid fluid). TenGSHui will be released in 2018, a companion paper is in preparation.

- Non-linear interaction of inertial modes
We have studied numerically the triad resonances of inertial modes in a spherical shell (geometrically similar to the Earth\'s core). These inertial modes are a manifestation of the Coriolis force acting in the fluid core. If the inner core rotates at a different angular speed compared to the mantle, then inertial modes can get excited, extracting energy from the differential rotation, and eventually lead to widespread turbulence thanks to a cascade of triad-wave interactions. This study is a follow-up to a previous experimental study that we performed using the spherical shell device at TU Cottbus (Hoff, Harlander & Triana, Physical Review F, vol 1, 043701, 2016). The present numerical study appeared in the May 2018 issue of the Journal of Fluid Mechanics (Barik, Triana, Hoff & Wicht, JFM, 843, 25 May 2018, 211-243).

- Instabilities, turbulence and dynamo generated by precession of a planet with an inner core.
When considering a planetary liquid layer, precession is generally present, driving flows, hydrodynamic instabilities and perhaps dynamos. However, there is no systematic study of these flows in the spherical shell geometry relevant for planets with an inner core, which makes difficult any extrapolation to planetary regimes. In this study, we have considered a large number of magneto-hydrodynamic simulations of precessing spherical shells, where all the parameters have been systematically varied.
We used this large simulations database to study the forced basic flows, the associated instabilities and the dynamo capability of these flows. For instance, we derived and validate an explicit analytical estimate of the viscous dissipation obtained in our simulations. We also propose theoretical onsets for instabilities in precessing spherical shells, showing that the parametric instabilities due to the outer boundary (the mantle) conical layer are controlling the stability in most cases. The present study has been submitted to Geophysical Journal International and is still under consideration.

- Solve dynamic equations with FENNEC
All of our studies involve the numerical resolution of systems of partial differential equations in near-spherical geometry. We have built a Mathematica package to automatically translate such set of equations and their boundary conditions into algebraic problems that can be solved on the computer. This package is built on top of TenGSHui (see Antonyâ€™s section). This was applied to the study of inertial modes of an inviscid fluid inside an ellipsoidal container and confronted to analytical solutions which are known in this simple case, thus confirming the validity of the approach (Rekier et al., 2018, under revision). FENNEC should be the object of a future publication accompanying its release.

- Data analysis
Interpreting variations in the magnitude and direction of the Earthâ€™s rotation vector, which brings together many diverse areas of study notably solid geophysics, geodesy, meteorology, etc. In this work package, we mainly focused on the nutation part in Earthâ€™s Orientation Parameters (EOPs) solutions. To study the observed nutation series, a priori nutation model is needed. An open-Source Python code has been developed and is now available on the project website \'rotanut.oma.be\', which can be used to compute the theoretical nutation values at any given timestamp based on the current Interna

Final results

The general objective of our science is advancing our understanding of the dynamic Earth system by quantifying our planetâ€™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.

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.