The displacement field accompanying the wobble/nutation of the Earth is conventionally represented by an infinite chain of toroidal and spheroidal vector spherical harmonics, coupled by rotation and ellipticity. Numerical solutions for the eigenperiods require truncation of that chain, and the standard approaches using the linear momentum description (LMD) of deformation during wobble/nutation have truncated it at very low degrees, usually degree 3 or 4, and at most degree 5. The effects of such heavy truncation on the computed eigenperiods have hardly been examined. We here investigate the truncation effects on the periods of the free wobble/nutation modes using a simplified Earth model consisting of a homogeneous incompressible inviscid liquid outer core with a rigid (but not fixed) inner core and mantle. A novel Galerkin method is implemented using a Clairaut coordinate system to solve the classic Poincar´e problem in the liquid core and, to close the problem, we use the Lagrangeanformulation of the Liouville equation for each of the solid parts of the Earth model. We find that, except for the free inner core nutation (FICN), the periods of the free rotational modes converge ratherquickly. The period of the tiltover mode (TOM) is found to excellent accuracy. The computed periods of the Chandler wobble (CW) and free core nutation (FCN) are nearly identical to the values cited in the literature for similar Earth models, but that for the inner core wobble (ICW) is slightly different. Truncation at low-degree harmonics causes the FICN period to fluctuate over a range as large as 90 sd, with different values at different truncation levels. For example, truncation at degree 6 gives a period of 752 sd (almost identical with the value cited in the literature for such an Earth model) but truncation at degree 24 is required to obtain convergence, and the resulting period is 746 1 sd, as more terms are included, with no guarantee that its proximity to earlier values is other than fortuitous. We conclude that the heavy truncation necessitated by the conventional LMD is unsatisfactory for the FICN.
The Earth’s outer core is a rotating ellipsoidal shell of compressible, stratified and self gravitating fluid. As such, in the treatment of geophysical problems a realistic model of this body needs to be considered. In this work, we consider compressible and stratified fluid core models with different stratification parameters, related to the local Brunt-Vaisala frequency, in order to study the effects of the core’s density stratification on the frequencies of some of the inertial-gravity modes of this body. The inertial-gravity modes of the core are free oscillations with periods longer than 12 hr. Historically, an incompressible and homogeneous fluid is considered to study these modes and analytical solutions are known for the frequencies and the displacement eigenfunctions of a spherical model. We show that for a compressible and stratified spherical core model the effects of non-neutral density stratification may be significant, and the frequencies of these modes may change from model to model....
If the reference state of a rotating and self gravitating fluid body is one of hydrostatic equilibrium then the figure of the body is a spheroid such that a cross sectional area parallel to the equatorial plane of the body is a circle while that parallel to a meridional plane is an ellipse. The effects of the fluid body’s flattened (spheroidal) figure is small on the frequencies of the body’s short-period (shorter than a few hours in the case of the Earth) normal modes. For the long-period normal modes, however, these effects must be considered. Furthermore, the body’s wobble and nutation modes owe their existence to its ellipsoidal figure. In the conventional approach to computing these frequencies, an orthogonal coordinate system is usually considered. It is then necessary to have the knowledge of the derivatives of the material properties of the body, such as the density and Lamé parameters, in order to include the effects of the ellipticity in the dynamical equations. In the available Earth models, however, these derivatives are not well defined. In order to minimize the effects of these derivatives in the treatment of the dynamical problems we use a non-orthogonal (Clairaut) coordinate system. Using this approach, we compute the frequencies and displacement eigenfunctions for some of the inertial modes of a realistic spheroidal model of the Earth’s fluid core and compare them to the known results for an Earth model with a homogeneous and incompressible fluid core.
We use the three potential description (3PD) to study the oscillatory dynamics of rotating stars. The 3PD scheme describes the exact linearized dynamics of rotating, self-gravitating, stratified, compressible and inviscid fluids. We compute the frequencies and the displacement eigenfunctions for some of the inertial and gravity modes of polytropic star models. The Coriolis force is the restoring force for the inertial modes while the gravity modes are induced by the buoyancy forces. We show that the frequencies of the inertial modes of these star models may be significantly different from those of a homogeneous and incompressible model, an approximation usually made to compute the frequencies of the inertial modes of rapidly rotating stars. We will also investigate the effects of rotation on the frequencies of the gravity modes of these models. In order to test the validity and the accuracy of the results from our approach, we compare the eigenfrequencies of some of the gravito-inertial modes of these models, computed using the 3PD, to those in the literature.
We propose a causal relationship between the creation of the giant impact basins on Mars by a large asteroid, ruptured when it entered the Roche limit, and the excitation of the Martian core dynamo. Our laboratory experiments indicate that the elliptical instability of the Martian core can be excited if the asteroid continually exerts tidal forces on Mars for ~20,000 years. Our numerical experiments suggest that the growth-time of the instability was 5,000–15,000 years when the asteroid was at a distance of 50,000–75,000 km. We demonstrate the stability of the orbital motion of an asteroid captured by Mars at a distance of 100,000 km in the presence of the Sun and Jupiter. We also present our results for the tidal interaction of the asteroid with Mars. An asteroidcaptured by Mars in prograde fashion can survive and excite the elliptical instability of the core for only a few million years, whereas a captured retrograde asteroid can excite the elliptical instability for hundreds of millions of years before colliding with Mars. The rate at which tidal energy dissipates in Mars during this period is over two orders of magnitude greater than the rate at which magnetic energy dissipates. If only 1% of the tidal energy dissipation is partitioned to the core, sufficient energy would be available to maintain the core dynamo. Accordingly, a retrograde asteroid is quite capable of exciting an elliptical instability in the Martian core, thus providing a candidate process todrive a core dynamo.
Computational methods are used to investigate the effects of fluid ompressibility on the frequencies of the inertial modes of the Earth’s fluid core. The 3PD (the three potential description) is applied to two models of compressible fluid spheres, for one of which the dilatation vanishes, in order to study these modes. We show that compressibility may have a significant effect on some of the modal frequencies. Using the shape of the displacement eigenfunctions of a fluid sphere we also infer which modes of a sphere may have counterparts in a spherical shell. We then give a few examples of the inertial modes of a compressible fluid shell proportional to the Earth’s fluid core.
The oscillatory dynamics of a rotating, self-gravitating, stratified, compressible, inviscid fluid body is simplified by an exact description in terms of three scalar fields which are constructed from the dilatation, and the perturbations in pressure and gravitational potential [Seyed-Mahmoud, B., 1994.Wobble/nutation of a rotating ellipsoidal Earth with liquid core: implementation of a new set of equations describing dynamics of rotating fluids M.Sc. Thesis, Memorial University of Newfoundland]. We test the method by applying it to compressible, but neutrally-stratified, models of the Earth’s liquid core, including a solid inner core, and compute the frequencies of some of the inertial modes. We conclude the method should be further exploited for astrophysical and geophysical normal mode computations.
If a circular flow of a contained rotating fluid is strained into an elliptical one in such a way that the elliptical streamlines preserve their figures in a laboratory reference frame, the flow becomes unstable. This type of instability is known to be excited as a result of coupling between a pair of inertial modes of the contained fluid by the applied strain. The present work is concerned with the elliptical instability of a thick, spherical, rotating fluid shell. We have conducted experiments and have observed the excitation of the elliptical instability as well as some of the inertial modes of a rotating fluid contained in the above mentioned geometry. The velocities of the fluid particles are captured by means of Digital Particle Imaging Velocimetry (DPIV). Plots of the velocity vectors of the inertial modes obtained experimentally are used to identify the modes predicted using theoretical methods. Such investigations are of geophysical interest, since tidal forcing might be sufficient to excite an elliptical instability in Earth’s fluid core as well as in other planetary interiors.
A dynamical model is proposed for the elliptical instability that has been reported by Aldridge et al. [Aldridge, K.D., Seyed-Mahmoud, B., Henderson, G.A., van Wijngaarden, W., 1997. Elliptical instability of the Earth’s fluid core. Phys. Earth Planet. Inter., 103, 365–374] in connection with recent experiments on an ellipsoidal shell of rotating fluid. The frequencies and growth rates of the instability are obtained numerically by means of a Galerkin method that is based upon the normal modes of the contained fluid. A finite-element method has been employed to approximately solve the ill-posed Poincare´ problem for the normal modes. The numerical results for a special case are compared with their analytical counterparts, and the agreement is to within 0.1% for shells of small ellipticity. Results are presented for other cases, including some where the boundary perturbation is allowed to rotate slowly with respect to the inertial frame. The conclusion is that such investigations are of geophysical interest, since tidal forcing might be sufficient to excite an elliptical instability of the fluid outer core of the Earth and thus contribute to the geomagnetic field.
The elliptical instability of a rotating fluid contained in a thick spherical shell has been excited in our laboratory by a tide-like perturbation of the flexible inner boundary. For an inviscid fluid, the growth rate of the instability is approximately proportional to the perturbation amplitude and the rotation rate. Development of the instability appears to be limited by the spherical outer surface and the relatively small perturbation applied over the inner surface. If the corresponding instability were excited in the Earth's fluid core by tidal forces, in the absence of dissipation the e-folding time for growth would be on the order of several thousand years. Although this time scale is similar to current estimates for the time needed for the geomagnetic field to undergo a reversal, the instability would grow at a rate equal to the difference between the ideal growth rate and the overall decay rate. The rates of viscous and electromagnetic damping are determined by material properties of the core fluid that are not well known. If elliptical instability plays a central role in geomagnetic reversals, upper limits on the viscosity and conductivity of the fluid core might be inferred.