The influence of rotation on the free oscillations of the Earth

1975 ◽  
Vol 279 (1292) ◽  
pp. 583-624 ◽  
Keyword(s):  
Rigid Body ◽  
Normal Mode ◽  
Body Force ◽  
Normal Modes ◽  
Earth Model ◽  
Static Response ◽  
The Earth ◽  
Rigid Body Modes ◽  
Effects Of Rotation ◽  
Zero Frequency

We pursue an abstract investigation of the theory of the infinitesimal free elasticgravitational oscillations of a fairly general rotating Earth model. By considering in some detail the transition to the non-rotating case, we are able to delineate certain of the intrinsic effects of rotation on the normal mode eigensolutions, and to show how profoundly rotation alters the fundamental mathematical and physical properties of these eigensolutions. In particular, we show that the displacement eigenfunctions of a rotating Earth model are not mutually orthogonal, and that the corresponding normal modes of oscillation cannot in general be represented by pure standing waves. We consider the excitation of the normal modes of oscillation of a rotating Earth model by a transient imposed body force distribution, and we show that the complex dynamical amplitude of each normal mode may, in many geophysical applications, be determined separately, in spite of the lack of orthogonality among the displacement eigenfunctions. The calculation of the associated static response after the decay of the normal modes of oscillation is, on the other hand, complicated considerably by the absence of orthogonality. We specifically examine the influence of rotation on the zero-frequency rigid body translational and rotational modes of any non-rotating Earth model, and show how to account for the corresponding rigid body modes of any rotating Earth model in excitation calculations.

scholarly journals Rigid-Earth Nutation Models

2000 ◽  
Vol 180 ◽  
pp. 190-195
Author(s):  
J. Souchay
Keyword(s):  
Transfer Function ◽  
Rigid Body ◽  
Earth Model ◽  
High Quality ◽  
Frequency Dependent ◽  
Relative Improvement ◽  
The Earth ◽  
Rigid Earth

AbstractDespite the fact that the main causes of the differences between the observed Earth nutation and that derived from analytical calculations come from geophysical effects associated with nonrigidity (core flattening, core-mantle interactions, oceans, etc…), efforts have been made recently to compute the nutation of the Earth when it is considered to be a rigid body, giving birth to several “rigid Earth nutation models.” The reason for these efforts is that any coefficient of nutation for a realistic Earth (including effects due to nonrigidity) is calculated starting from a coefficient for a rigid-Earth model, using a frequency-dependent transfer function. Therefore it is important to achieve high quality in the determination of rigid-Earth nutation coefficients, in order to isolate the nonrigid effects still not well-modeled.After reviewing various rigid-Earth nutation models which have been established recently and their relative improvement with respect to older ones, we discuss their specifics and their degree of agreement.

Backus-Gilbert style inversions for mantle anisotropy using normal mode data 

2021 ◽  
Author(s):  
Federica Restelli ◽  
Paula Koelemeijer ◽  
Christophe Zaroli
Keyword(s):  
Normal Mode ◽  
Ad Hoc ◽  
Seismic Anisotropy ◽  
Normal Modes ◽  
Least Square ◽  
Mantle Flow ◽  
Tomographic Images ◽  
The Earth ◽  
Initial Work ◽  
Mantle Anisotropy

<p>Seismic tomography is essential for imaging the Earth’s interior in order to better understand the dynamic processes at work. However, robust physical interpretation of tomographic images remain difficult as the inverse problem is under-determined, model amplitudes are biased and uncertainties are usually not quantified.</p><p>Commonly-used techniques, such as damped least-square inversions, break the non-uniqueness of the model solution by adding a subjective, ad hoc, regularization, which can lead to biased amplitudes and potential physical misinterpretations. The SOLA method (Zaroli, 2016; Zaroli et al., 2017), based on a Backus-Gilbert approach, removes the non-uniquess by averaging, rather than introducing a subjective regularization. The method explicitly constrains the amplitudes to be unbiased and the computation of the model resolution and uncertainty is inherent and efficient. Instead of aiming to minimize the data fit, the SOLA approach aims to minimize the size of the averaging volume <!-- Think it is clear enough without the extra sentence – as averaging is mentioned before -->and the associated uncertainties.</p><p>We aim to build a new tomographic model of the Earth’s mantle using the SOLA method. We focus our observations on normal mode data, the standing waves of the Earth observed after very large earthquakes, which are not affected by an uneven data distribution. As normal modes are sensitive to multiple seismic parameters, we treat the sensitivity to different parameters as so called “3D noise” within the SOLA framework. We are specifically interested in constraining seismic anisotropy, which provides more direct information on mantle flow.</p><p>Here, we report on some forward modelling results, fundamental to understanding normal mode sensitivity to seismic anisotropy at different depths and identifying which modes to focus on during inversions. We also show our initial work towards building a new tomography model, including the calculation of 3D noise and target kernels.</p><p> </p>

Uniqueness in the inversion of inaccurate gross Earth data

1970 ◽  
Vol 266 (1173) ◽  
pp. 123-192 ◽  
Keyword(s):  
Normal Mode ◽  
Normal Modes ◽  
Variance Matrix ◽  
The Earth ◽  
Mass Moment ◽  
Frequency Independent ◽  
Finite Set ◽  
Mass Moment Of Inertia ◽  
Limits Of Error ◽  
Local Average

A gross Earth datum is a single measurable number describing some property of the whole Earth, such as mass, moment of inertia, or the frequency of oscillation of some identified elastic-gravitational normal mode. We suppose that a finite set G of gross Earth data has been measured, that the measurements are inaccurate, and that the variance matrix of the errors of measurement can be estimated. We show that some such sets G of measurements determine the structure of the Earth within certain limits of error except for fine-scale detail. That is, from some setsG it is possible to compute localized averages of the Earth structure at various depths. These localized averages will be slightly in error, and their errors will be larger as their resolving lengths are shortened. We show how to determine whether a given set G of measured gross Earth data permits such a construction of localized averages, and, if so, how to find the shortest length scale over which G gives a local average structure at a particular depth if the variance of the error in computing that local average from G is to be less than a specified amount. We apply the general theory to the linear problem of finding the depth variation of a frequency-independent local elastic dissipation ( Q ) from the observed damping rates of a finite number of normal modes. We also apply the theory to the nonlinear problem of finding density against depth from the total mass, moment and normal-mode frequencies, in case the compressional and shear velocities are known.

FrosPy: A Modular Python Toolbox for Normal Mode Seismology

2022 ◽  
Author(s):  
Simon Schneider ◽  
Sujania Talavera-Soza ◽  
Lisanne Jagt ◽  
Arwen Deuss
Keyword(s):  
Normal Mode ◽  
Free Oscillation ◽  
Large Time ◽  
Reference Model ◽  
Normal Modes ◽  
Earth Model ◽  
Splitting Function ◽  
Quality Factor Q ◽  
Earth’S Free Oscillation ◽  
Q Values

Abstract We present free oscillations Python (FrosPy), a modular Python toolbox for normal mode seismology, incorporating several Python core classes that can easily be used and be included in larger Python programs. FrosPy is freely available and open source online. It provides tools to facilitate pre- and postprocessing of seismic normal mode spectra, including editing large time series and plotting spectra in the frequency domain. It also contains a comprehensive database of center frequencies and quality factor (Q) values based on 1D reference model preliminary reference Earth model for all normal modes up to 10 mHz and a collection of published measurements of center frequencies, Q values, and splitting function (or structure) coefficients. FrosPy provides the tools to visualize and convert different formats of splitting function coefficients and plot these as maps. By giving the means of using and comparing normal mode spectra and splitting function measurements, FrosPy also aims to encourage seismologists and geophysicists to learn about normal mode seismology and the study of the Earth’s free oscillation spectra and to incorporate them into their own research or use them for educational purposes.

scholarly journals A Normal Mode Study of Wobble and Nutation

1980 ◽  
Vol 78 ◽  
pp. 195-202
Author(s):  
Martin L. Smith
Keyword(s):  
Rigid Body ◽  
Normal Mode ◽  
Chandler Wobble ◽  
The Earth ◽  
Rotating Bodies ◽  
Mode Study

The observed eigenperiod of the Chandler Wobble is about 435.2 sidereal days while the theoretical eigenperiod of a rigid body having the same composition and geometry as the Earth is about 305 days. The attempt to reconcile these two numbers has led scientists to study theoretically the free wobble and nutation of various classes of rotating bodies.

Acoustic Green's Function Approximations

1998 ◽  
Vol 06 (04) ◽  
pp. 435-452 ◽  
Author(s):  
Robert P. Gilbert ◽  
Zhongyan Lin ◽  
Klaus Hackl
Keyword(s):  
Inverse Problem ◽  
Normal Mode ◽  
Eigenvalue Problems ◽  
Normal Modes ◽  
Modal Parameters ◽  
Mindlin Plate ◽  
Ocean Bottom ◽  
Hankel Transformation ◽  
Poroelastic Materials ◽  
Function Approximations

Normal-mode expansions for Green's functions are derived for ocean–bottom systems. The bottom is modeled by Kirchhoff and Reissner–Mindlin plate theories for elastic and poroelastic materials. The resulting eigenvalue problems for the modal parameters are investigated. Normal modes are calculated by Hankel transformation of the underlying equations. Finally, the relation to the inverse problem is outlined.

Experimental Study for Extraction of Normal Modes

1995 ◽  
Author(s):  
S. Y. Chen ◽  
M. S. Ju ◽  
Y. G. Tsuei
Keyword(s):  
Frequency Response ◽  
Normal Mode ◽  
Normal Modes ◽  
Measurement Data ◽  
Mode Shapes ◽  
Complex Frequency ◽  
Response Functions ◽  
Frequency Response Functions ◽  
Incomplete System ◽  
Coupled Structures

Abstract A frequency-domain technique to extract the normal mode from the measurement data for highly coupled structures is developed. The relation between the complex frequency response functions and the normal frequency response functions is derived. An algorithm is developed to calculate the normal modes from the complex frequency response functions. In this algorithm, only the magnitude and phase data at the undamped natural frequencies are utilized to extract the normal mode shapes. In addition, the developed technique is independent of the damping types. It is only dependent on the model of analysis. Two experimental examples are employed to illustrate the applicability of the technique. The effects due to different measurement locations are addressed. The results indicate that this technique can successfully extract the normal modes from the noisy frequency response functions of a highly coupled incomplete system.

A Rapid Approach for Calculating the Damped Eigenvalues of a Gas Turbine on a Minicomputer: Theory

1984 ◽  
Vol 106 (2) ◽  
pp. 239-249 ◽  
Author(s):  
E. J. Gunter ◽  
R. R. Humphris ◽  
H. Springer
Keyword(s):  
Gas Turbine ◽  
Characteristic Polynomial ◽  
Normal Modes ◽  
Complex Matrix ◽  
Large Computer ◽  
Mainframe Computer ◽  
The Matrix ◽  
Rigid Body Modes ◽  
Transfer Method ◽  
Matrix Transfer Method

The calculation of the damped eigenvalues of a large multistation gas turbine by the complex matrix transfer procedure may encounter numerical difficulties, even on a large computer due to numerical round-off errors. In this paper, a procedure is presented in which the damped eigenvalues may be rapidly and accurately calculated on a minicomputer with accuracy which rivals that of a mainframe computer using the matrix transfer method. The method presented in this paper is based upon the use of constrained normal modes plus the rigid body modes in order to generate the characteristic polynomial of the system. The constrained undamped modes, using the matrix transfer process with scaling, may be very accurately calculated for a multistation turbine on a minicomputer. In this paper, a five station rotor is evaluated to demonstrate the procedure. A method is presented in which the characteristic polynomial may be automatically generated by Leverrier’s algorithm. The characteristic polynomial may be directly solved or the coefficients of the polynomial may be examined by the Routh criteria to determine stability. The method is accurate and easy to implement on a 16 bit minicomputer.

scholarly journals Magneto-rotatory compressible couple-stress fluid heated from below in porous medium

2016 ◽  
Vol 38 (1) ◽  
pp. 55-63
Author(s):  
Chander Bhan Mehta
Keyword(s):  
Magnetic Field ◽  
Porous Medium ◽  
Couple Stress ◽  
Normal Modes ◽  
Sufficient Conditions ◽  
Couple Stress Fluid ◽  
Onset Of Convection ◽  
Medium Permeability ◽  
Effects Of Rotation ◽  
The Magnetic Field

Abstract The study is aimed at analysing thermal convection in a compressible couple stress fluid in a porous medium in the presence of rotation and magnetic field. After linearizing the relevant equations, the perturbation equations are analysed in terms of normal modes. A dispersion relation governing the effects of rotation, magnetic field, couple stress parameter and medium permeability have been examined. For a stationary convection, the rotation postpones the onset of convection in a couple stress fluid heated from below in a porous medium in the presence of a magnetic field. Whereas, the magnetic field and couple stress postpones and hastens the onset of convection in the presence of rotation and the medium permeability hastens and postpones the onset of convection with conditions on Taylor number. Further the oscillatory modes are introduced due to the presence of rotation and the magnetic field which were non-existent in their absence, and hence the principle of exchange stands valid. The sufficient conditions for nonexistence of over stability are also obtained.

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