## Periodic Reporting for period 1 - RotaNut (Rotation and Nutation of a wobbly Earth)

**Reporting period:**2015-09-01

**to**2017-02-28

## Summary of the context and overall objectives of the project

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 – 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 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 adopted model is based on the idea that 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 Earth’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 of a liquid core inside the Earth and all the coupling mechanisms at the core-mantle boundary play indeed an important role in nutation amplitudes. An everyday example of the influence of the physical state of the interior on the rotation is that raw (liquid) and cooked (solid) eggs rotate differently. For the Earth, the identified mechanisms considered in the previous nutation model involve the flattening of the core and the fluid pressure and gravitational effect on that flattened boundary as well as a simple dipole and uniform magnetic field effect, because of their role in transferring angular momentum from the core to the mantle. By now, we have understood that there are important contributions from other components of the magnetic field, and that the dynamics in the liquid core and the presence of a topography at the core-mantle boundary at the kilometre level play an important role as well.

Why is it important for society?

The rotation of the Earth has long been used as a measure of time, and the stars as reference points to determine travellers’ whereabouts on the globe. Today, inexpensive global positioning systems (GPS) receive data sent by orbiting satellites and therewith can pinpoint the location of a person or a vehicle to within just a few metres and can provide the time from the GPS clock at below the tenth of a millisecond. Precise timescales are provided using atomic clocks and precise positioning is determined using geodetic techniques such as GPS grounded on two reference frames: the terrestrial frame, fixed relative to the Earth and rotating synchronously with the planet, and the celestial frame, which is immobile in space, where the artificial satellites such as those of GPS are moving. For both public GPS and precise scientific positioning or timing one definitely needs to know precisely the link between the celestial frame and the terrestrial frame, i.e. the Earth orientation in space and its precise rotation speed.

What are the overall objectives?

The main objective of the project RotaNut is to significantly improve the model for the orientation of the spin axis to just the sub-centimetre, which has never previously been achieved and which is the aim of all existing decadal surveys in geodesy. This will be extremely helpful to European and international satellite missions and GNSS (Global Navigation Satellite System, such as GPS or its European equivalent Galileo) positioning. At the same time, it will allow scientists to learn much more about the interior of the Earth such as the predominant coupling mechanisms at the core boundaries (inner core boundary and core-mantle boundary) at nutation timescale and the amplitude of the magnetic field at these boundaries, which are not directly observable from the Earth surface (as only a part of the field is observable above the insulating mantle).

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 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 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 adopted model is based on the idea that 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 Earth’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 of a liquid core inside the Earth and all the coupling mechanisms at the core-mantle boundary play indeed an important role in nutation amplitudes. An everyday example of the influence of the physical state of the interior on the rotation is that raw (liquid) and cooked (solid) eggs rotate differently. For the Earth, the identified mechanisms considered in the previous nutation model involve the flattening of the core and the fluid pressure and gravitational effect on that flattened boundary as well as a simple dipole and uniform magnetic field effect, because of their role in transferring angular momentum from the core to the mantle. By now, we have understood that there are important contributions from other components of the magnetic field, and that the dynamics in the liquid core and the presence of a topography at the core-mantle boundary at the kilometre level play an important role as well.

Why is it important for society?

The rotation of the Earth has long been used as a measure of time, and the stars as reference points to determine travellers’ whereabouts on the globe. Today, inexpensive global positioning systems (GPS) receive data sent by orbiting satellites and therewith can pinpoint the location of a person or a vehicle to within just a few metres and can provide the time from the GPS clock at below the tenth of a millisecond. Precise timescales are provided using atomic clocks and precise positioning is determined using geodetic techniques such as GPS grounded on two reference frames: the terrestrial frame, fixed relative to the Earth and rotating synchronously with the planet, and the celestial frame, which is immobile in space, where the artificial satellites such as those of GPS are moving. For both public GPS and precise scientific positioning or timing one definitely needs to know precisely the link between the celestial frame and the terrestrial frame, i.e. the Earth orientation in space and its precise rotation speed.

What are the overall objectives?

The main objective of the project RotaNut is to significantly improve the model for the orientation of the spin axis to just the sub-centimetre, which has never previously been achieved and which is the aim of all existing decadal surveys in geodesy. This will be extremely helpful to European and international satellite missions and GNSS (Global Navigation Satellite System, such as GPS or its European equivalent Galileo) positioning. At the same time, it will allow scientists to learn much more about the interior of the Earth such as the predominant coupling mechanisms at the core boundaries (inner core boundary and core-mantle boundary) at nutation timescale and the amplitude of the magnetic field at these boundaries, which are not directly observable from the Earth surface (as only a part of the field is observable above the insulating mantle).

## Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

For this first period, we have addressed the following tasks. We also thereafter mention the results obtained.

1. Computation of nutation and comparison between theory and observations; determination of basic Earth parameters – P. Zhu

P. Zhu has developed codes for nutation computation (version v01), including the following model: (1) K77 precession/ IAU1980 nutation model (Very helpful to evaluate the long period nutation terms), (2) IAU2000a nutation and IAU2006 precession model (Reference for our analysis), (3) MHB2000 nutation model.

There are several global VLBI solutions available from different data centers located in Europe and the United States of America. We began our analysis using the global VLBI solutions: IERS-EOP08-C04, a product of International Earth Rotation Service. P. Zhu has developed codes for determination of the basic Earth interior parameters from using VLBI observation (version v01). He then applied this to different VLBI series.

We first adjusted the 106 nutation terms of the IAU1980 nutation model, then all the largest terms. We thus developed the following code: (4) a least-square code that is fitting all the nutation terms >2 microarcsecond terms. We applied this code to 35 years of VLBI observations. The re-estimated amplitudes of nutation residuals remain large with respect to the previously adopted nutation series (computed from the above-mentioned codes). Among them, the most significant deviations compared to MHB2000 nutation series are for the long period nutations at 18.6 years and 9.3 years as well as for the prograde nutation at 182.6 days. This last nutation can also be strongly perturbed by the atmospheric loading.

Further, the free mode included in the observation had to be taken out of the data. We developed a code to be applied before the computation of the nutation residual amplitudes, computing: (5) empirical Free Core Nutation (FCN) model.

In order to further analyse the residuals, we developed: (6) a wavelets-tool to analyse nutation residuals. From these nutation residual amplitudes, it is possible to derive now the Basic Earth Parameters (BEP), which are related to the Earth interior unknown feature that we want to determine as an ultimate objective. (7) A least-square fit to evaluate the BEP from the nutation amplitudes, (8) a statistic model to evaluate the EOP and parameter uncertainties and the nutation model accuracy, (9) using Koot’s code, a Bayesian approach for evaluating the BEP from the nutation amplitudes.

2. Mixing of rotational modes with gravito-inertial and hydromagnetic modes – J. Rekier, S. A. Triana, and A. Trinh

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.

The Earth is a relatively fast-spinning body, and the fluid character of the outer core allows for the possibility of inertial waves, i.e. waves that are restored by the Coriolis force. There are many aspects of these waves that remain poorly understood, even in ‘simple’ cases involving homogenous and incompressible fluids. Inertial eigenmodes exhibit internal shear layers that emanate from the so-called critical latitudes. They propagate throughout the fluid and become thinner as the Ekman number (a dimensionless quantity representing the ratio of viscous to Coriolis forces the fluid) decreases. Although the viscous dissipation associated with these layers decreases with the Ekman number, the Ohmic dissipation increases in these localized shear flow regions. The presence of inertial waves can then influence dramatically the flow dynamics, particularly at the extremely low Ekman numbers associated with the Earth and other planetary cores.

We believe that taking into account a more realistic picture of the liquid layer, which is in general roughly approximated, could modify the eigenfrequencies of the system. These inertial oscillations can couple to the rotational modes in ways that are still not well understood. This possibility was suggested by Rogister & Valette (2009) and may help to explain anomalous phase jumps of the observed free core nutation or coupling constants at the core boundaries as determined from the nutation amplitudes.

We have designed a novel approach to compute these global eigenfrequencies and the corresponding eigenmodes. The resulting equations are composed of the Navier-Stokes equations for the fluid coupled to the Liouville equations for the mantle and the solid inner core.

In a frame attached to the mantle, the time-dependent motion of the boundary results in a fictitious force, known as the Poincaré force, which is included in the Navier-Stokes equation describing the fluid dynamical part, in addition to the other familiar fictitious forces, Coriolis and centrifugal.

These equations are coupled through various torques at the inner core boundary and core-mantle boundary. We focused first on the pressure torque arising from pressure coupling due to the non-spherical shape of the interfaces, since it is the dominant contribution to the total torque. We also take into account the smaller viscous torque. Topographic and electro-magnetic torques are also important and will be considered at a later stage.

Given the wide range of problems showing up in the other projects mentioned in this report involving the resolution of systems of differential equations, this project aims at putting in place the numerical tools to help in these tasks.

We developed two different codes to determine the eigenfrequencies of the Earth’s core.

(10) The first code (SpODES for Spectral ODE Solver) is built on a spectral decomposition in the three dimensions and is designed with focus on adaptability and flexibility.

(11) The second code uses finite differences in the radial direction and is designed with focus on computational performance, which allows exploring low-viscosity settings.

Additional physical mechanisms can be easily included in the first code for exploration purposes, and then implemented in the second code for systematic exploration of the parameter space.

3. Automate the writing of tensor equations in celestial bodies – A. Trinh

To numerically implement the tensor equations governing the global dynamics of the Earth, we use expansions in generalized spherical harmonics. These require lengthy algebraic calculations that should better not be treated by hand.

(12) A. Trinh developed TenGSHui, a Mathematica package to handle tensor equations in slightly aspherical configurations. Three systems of coordinates can now be used: spherical coordinates, Clairaut coordinates, and (for scalar fields only) oblate spheroidal coordinates. This allowed to validate the convergence of approximate series expansions (in powers of the flattening) in spherical coordinates against exact expressions in oblate spheroidal coordinates and ellipsoidal coordinates (for the Poisson equation), and in bi-spheroidal coordinates and bi-ellipsoidal coordinates (for the Poincaré equation). Tensor calculus over surfaces, required in the boundary conditions of non-hydrostatic configurations, is now also implemented. Finally, computational demands are now significantly reduced, making it possible to truncate equations at much higher spherical/spheroidal harmonic degree.

An application of the TenGSHui software was performed and published in Geophysical Research Letters (Beuthe et al., 2016).

4. Inner core differential rotation – Inertial wave instability observed in experiments using the same geometrical configuration as the Earth’s core (spherical-Couette) – S.A. Triana

Rotating fluid experiments involving a differentially rotating inner core have revealed that inertial modes are excited and draw energy from the shear flow. The mechanism that triggers this inertial wave instability is still not understood. The angular momentum is redistributed in the flow via non-linear interactions, which under some circumstances can also lead to broadband turbulence. The aim of this research is to understand the inertial wave excitation in detail using experiments and numerics.

The inertial wave instability observed in experiments using the same geometrical configuration as the Earth’s core (spherical-Couette), appears to be much more complex than initially observed in the past. S.A. Triana and collaborators, using the rotating spherical-Couette device at BTU-Cottbus (Germany), have determined that the critical Rossby number necessary for this instability appears to be as low as 10^-6, which is about the same order of magnitude as the Rossby number typical of the Earth’s fluid outer core. It is then plausible that the inertial wave instability participates in the turbulence generation mechanism in the core. The results of this research have been published in the journal Physical Review Fluids (Hoff, Harlander and Triana 2016).

5. Influence of core dynamics on the magnetic field – R. Laguerre

The Earth’s magnetic field is generated by the so-called dynamo effect in the liquid core of the planet. Complex movements of the conducting fluid inside the core, coming from combined action of several forcing, sustain electrical currents and give rise to a large-scale magnetic field. The preferred mechanism for dynamo action inside the Earth used to be, either thermal convection due to strong temperature gradient between the inner core boundary (ICB) and the core mantle boundary (CMB) or compositional convection coming from the crystallization of the liquid iron at the ICB followed by the liberation of light elements.

The existence of a convective dynamo is strongly constrained by the values of thermal and electrical conductivities. Recent results seem to favor higher values of these conductivities than previously thought which would imply two things: first, much less power is available to sustain thermal convection and second, the corresponding age of the inner core would be around 1.5 billion years instead of 4 billion years as usually considered. If these recent measurements were confirmed, these effects would have a dramatic impact on our knowledge of the history of the Earth’s magnetic field. Indeed, the field being 4.5 billion years old, it would imply the existence of a long period without inner core during which the dynamo action would have to be sustained by mechanisms other than compositional convection. Other mechanisms, encountered in the planetary context, have shown their ability to sustain a dynamo effect. These include tidal forcing and precession forcing. We focus on the study of the ability of precessional flow to give rise to dynamo action.

Our objectives are to improve our understanding of instabilities generated in the liquid core by the precession of the Earth, the transition between these instabilities and turbulence and the ability of the resulting flows to sustain magnetic field through dynamo effect. More precisely, we want to characterize the influence of the solid inner on the threshold of the hydrodynamic instabilities, on the possibility of inverse cascades of energy in the turbulent regime and on the topology of the magnetic field (if sustained).

6. Effect of topographic features of the Core-Mantle Boundary – R. Laguerre, J. Rekier, A. Trinh and S.A. Triana

It is well known that the viscous (Ekman) layer of a fluid rotating inside a cavity breaks down at some critical latitudes leading to increased dissipation inside the fluid as an effect of spawning shear layers. In the RotaNut project, we want to investigate the possibility that these shear layers might also spawn from actual topographic features at the CMB that we expect to be present in the Earth.

Another approach to the core-mantle coupling problem involves the study of localized small-scale topography. This study has been initiated by Raphael Laguerre and Santiago Triana using the COMSOL Multiphysics Software to introduce small “bumps” on the fluid bounding surface. Given the speed limitations of the modelling software, these topographical features are restricted to be axisymmetric, i.e. they resemble a mountain ridge that circles the CMB azimuthally. This study is still in its initial stages.

1. Computation of nutation and comparison between theory and observations; determination of basic Earth parameters – P. Zhu

P. Zhu has developed codes for nutation computation (version v01), including the following model: (1) K77 precession/ IAU1980 nutation model (Very helpful to evaluate the long period nutation terms), (2) IAU2000a nutation and IAU2006 precession model (Reference for our analysis), (3) MHB2000 nutation model.

There are several global VLBI solutions available from different data centers located in Europe and the United States of America. We began our analysis using the global VLBI solutions: IERS-EOP08-C04, a product of International Earth Rotation Service. P. Zhu has developed codes for determination of the basic Earth interior parameters from using VLBI observation (version v01). He then applied this to different VLBI series.

We first adjusted the 106 nutation terms of the IAU1980 nutation model, then all the largest terms. We thus developed the following code: (4) a least-square code that is fitting all the nutation terms >2 microarcsecond terms. We applied this code to 35 years of VLBI observations. The re-estimated amplitudes of nutation residuals remain large with respect to the previously adopted nutation series (computed from the above-mentioned codes). Among them, the most significant deviations compared to MHB2000 nutation series are for the long period nutations at 18.6 years and 9.3 years as well as for the prograde nutation at 182.6 days. This last nutation can also be strongly perturbed by the atmospheric loading.

Further, the free mode included in the observation had to be taken out of the data. We developed a code to be applied before the computation of the nutation residual amplitudes, computing: (5) empirical Free Core Nutation (FCN) model.

In order to further analyse the residuals, we developed: (6) a wavelets-tool to analyse nutation residuals. From these nutation residual amplitudes, it is possible to derive now the Basic Earth Parameters (BEP), which are related to the Earth interior unknown feature that we want to determine as an ultimate objective. (7) A least-square fit to evaluate the BEP from the nutation amplitudes, (8) a statistic model to evaluate the EOP and parameter uncertainties and the nutation model accuracy, (9) using Koot’s code, a Bayesian approach for evaluating the BEP from the nutation amplitudes.

2. Mixing of rotational modes with gravito-inertial and hydromagnetic modes – J. Rekier, S. A. Triana, and A. Trinh

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.

The Earth is a relatively fast-spinning body, and the fluid character of the outer core allows for the possibility of inertial waves, i.e. waves that are restored by the Coriolis force. There are many aspects of these waves that remain poorly understood, even in ‘simple’ cases involving homogenous and incompressible fluids. Inertial eigenmodes exhibit internal shear layers that emanate from the so-called critical latitudes. They propagate throughout the fluid and become thinner as the Ekman number (a dimensionless quantity representing the ratio of viscous to Coriolis forces the fluid) decreases. Although the viscous dissipation associated with these layers decreases with the Ekman number, the Ohmic dissipation increases in these localized shear flow regions. The presence of inertial waves can then influence dramatically the flow dynamics, particularly at the extremely low Ekman numbers associated with the Earth and other planetary cores.

We believe that taking into account a more realistic picture of the liquid layer, which is in general roughly approximated, could modify the eigenfrequencies of the system. These inertial oscillations can couple to the rotational modes in ways that are still not well understood. This possibility was suggested by Rogister & Valette (2009) and may help to explain anomalous phase jumps of the observed free core nutation or coupling constants at the core boundaries as determined from the nutation amplitudes.

We have designed a novel approach to compute these global eigenfrequencies and the corresponding eigenmodes. The resulting equations are composed of the Navier-Stokes equations for the fluid coupled to the Liouville equations for the mantle and the solid inner core.

In a frame attached to the mantle, the time-dependent motion of the boundary results in a fictitious force, known as the Poincaré force, which is included in the Navier-Stokes equation describing the fluid dynamical part, in addition to the other familiar fictitious forces, Coriolis and centrifugal.

These equations are coupled through various torques at the inner core boundary and core-mantle boundary. We focused first on the pressure torque arising from pressure coupling due to the non-spherical shape of the interfaces, since it is the dominant contribution to the total torque. We also take into account the smaller viscous torque. Topographic and electro-magnetic torques are also important and will be considered at a later stage.

Given the wide range of problems showing up in the other projects mentioned in this report involving the resolution of systems of differential equations, this project aims at putting in place the numerical tools to help in these tasks.

We developed two different codes to determine the eigenfrequencies of the Earth’s core.

(10) The first code (SpODES for Spectral ODE Solver) is built on a spectral decomposition in the three dimensions and is designed with focus on adaptability and flexibility.

(11) The second code uses finite differences in the radial direction and is designed with focus on computational performance, which allows exploring low-viscosity settings.

Additional physical mechanisms can be easily included in the first code for exploration purposes, and then implemented in the second code for systematic exploration of the parameter space.

3. Automate the writing of tensor equations in celestial bodies – A. Trinh

To numerically implement the tensor equations governing the global dynamics of the Earth, we use expansions in generalized spherical harmonics. These require lengthy algebraic calculations that should better not be treated by hand.

(12) A. Trinh developed TenGSHui, a Mathematica package to handle tensor equations in slightly aspherical configurations. Three systems of coordinates can now be used: spherical coordinates, Clairaut coordinates, and (for scalar fields only) oblate spheroidal coordinates. This allowed to validate the convergence of approximate series expansions (in powers of the flattening) in spherical coordinates against exact expressions in oblate spheroidal coordinates and ellipsoidal coordinates (for the Poisson equation), and in bi-spheroidal coordinates and bi-ellipsoidal coordinates (for the Poincaré equation). Tensor calculus over surfaces, required in the boundary conditions of non-hydrostatic configurations, is now also implemented. Finally, computational demands are now significantly reduced, making it possible to truncate equations at much higher spherical/spheroidal harmonic degree.

An application of the TenGSHui software was performed and published in Geophysical Research Letters (Beuthe et al., 2016).

4. Inner core differential rotation – Inertial wave instability observed in experiments using the same geometrical configuration as the Earth’s core (spherical-Couette) – S.A. Triana

Rotating fluid experiments involving a differentially rotating inner core have revealed that inertial modes are excited and draw energy from the shear flow. The mechanism that triggers this inertial wave instability is still not understood. The angular momentum is redistributed in the flow via non-linear interactions, which under some circumstances can also lead to broadband turbulence. The aim of this research is to understand the inertial wave excitation in detail using experiments and numerics.

The inertial wave instability observed in experiments using the same geometrical configuration as the Earth’s core (spherical-Couette), appears to be much more complex than initially observed in the past. S.A. Triana and collaborators, using the rotating spherical-Couette device at BTU-Cottbus (Germany), have determined that the critical Rossby number necessary for this instability appears to be as low as 10^-6, which is about the same order of magnitude as the Rossby number typical of the Earth’s fluid outer core. It is then plausible that the inertial wave instability participates in the turbulence generation mechanism in the core. The results of this research have been published in the journal Physical Review Fluids (Hoff, Harlander and Triana 2016).

5. Influence of core dynamics on the magnetic field – R. Laguerre

The Earth’s magnetic field is generated by the so-called dynamo effect in the liquid core of the planet. Complex movements of the conducting fluid inside the core, coming from combined action of several forcing, sustain electrical currents and give rise to a large-scale magnetic field. The preferred mechanism for dynamo action inside the Earth used to be, either thermal convection due to strong temperature gradient between the inner core boundary (ICB) and the core mantle boundary (CMB) or compositional convection coming from the crystallization of the liquid iron at the ICB followed by the liberation of light elements.

The existence of a convective dynamo is strongly constrained by the values of thermal and electrical conductivities. Recent results seem to favor higher values of these conductivities than previously thought which would imply two things: first, much less power is available to sustain thermal convection and second, the corresponding age of the inner core would be around 1.5 billion years instead of 4 billion years as usually considered. If these recent measurements were confirmed, these effects would have a dramatic impact on our knowledge of the history of the Earth’s magnetic field. Indeed, the field being 4.5 billion years old, it would imply the existence of a long period without inner core during which the dynamo action would have to be sustained by mechanisms other than compositional convection. Other mechanisms, encountered in the planetary context, have shown their ability to sustain a dynamo effect. These include tidal forcing and precession forcing. We focus on the study of the ability of precessional flow to give rise to dynamo action.

Our objectives are to improve our understanding of instabilities generated in the liquid core by the precession of the Earth, the transition between these instabilities and turbulence and the ability of the resulting flows to sustain magnetic field through dynamo effect. More precisely, we want to characterize the influence of the solid inner on the threshold of the hydrodynamic instabilities, on the possibility of inverse cascades of energy in the turbulent regime and on the topology of the magnetic field (if sustained).

6. Effect of topographic features of the Core-Mantle Boundary – R. Laguerre, J. Rekier, A. Trinh and S.A. Triana

It is well known that the viscous (Ekman) layer of a fluid rotating inside a cavity breaks down at some critical latitudes leading to increased dissipation inside the fluid as an effect of spawning shear layers. In the RotaNut project, we want to investigate the possibility that these shear layers might also spawn from actual topographic features at the CMB that we expect to be present in the Earth.

Another approach to the core-mantle coupling problem involves the study of localized small-scale topography. This study has been initiated by Raphael Laguerre and Santiago Triana using the COMSOL Multiphysics Software to introduce small “bumps” on the fluid bounding surface. Given the speed limitations of the modelling software, these topographical features are restricted to be axisymmetric, i.e. they resemble a mountain ridge that circles the CMB azimuthally. This study is still in its initial stages.

## Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

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.

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.

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.