Rotational Multiplets in the Spectrum of the Earth

Pekeris, C. L.; Alterman, Z.; Jarosch, H.

doi:10.1103/physrev.122.1692

Cited by 79 publications

(28 citation statements)

References 6 publications

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“…In his article 25 , Mehl only reports on the l = 0 (dotted red line) and l = 1 (dashed red line) cases. The mode crossing we observe occurs for large enough negative value of γ (2) nlm and was therefore not present in his results. Our implementation of perturbation theory is in very good agreement with both Mehl's original results 25 and the finite-element numerical calculations, showing that perturbation theory up to second-order is sufficient for mode identification in ZoRo within our frequency range.…”

Section: B Second-order Ellipticity Perturbationmentioning

confidence: 54%

Acoustic spectra of a gas-filled rotating spheroid

Cébron

Nataf

et al. 2020

European Journal of Mechanics - B/Fluids

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The acoustic spectrum of a gas-filled resonating cavity can be used to indirectly probe its internal velocity field. This unconventional velocimetry method is particularly interesting for opaque fluid or rapidly rotating flows, which cannot be imaged with standard methods. This requires to (i) identify a large enough number of acoustic modes, (ii) accurately measure their frequencies, and (iii) compare with theoretical synthetic spectra. Relying on a dedicated experiment, an air-filled rotating spheroid of moderate ellipticity, our study addresses these three challenges. To do so, we use a comprehensive theoretical framework, together with finite-element calculations, and consider symmetry arguments. We show that the effects of the Coriolis force can be successfully retrieved through our acoustic measurements, providing the first experimental measurements of the rotational splitting (or Ledoux) coefficients for a large collection of modes. Our results pave the way for the modal acoustic velocimetry to be a robust, versatile, and non-intrusive method for mapping large-scale flows.Pages: 1-14

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Section: B Second-order Ellipticity Perturbationmentioning

confidence: 54%

Acoustic spectra of a gas-filled rotating spheroid

Cébron

Nataf

et al. 2020

European Journal of Mechanics - B/Fluids

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“…Recordings of this event also provided the first observational evidence for splitting of the lowest frequency modes. This splitting was attributed to the Earth's rotation by Backus and Gilbert [1961] and Pekeris et al [1961]. Splitting effect due to ellipticity was introduced later on by Usami and Satô [1962] and Dahlen [1968].…”

Section: Introductionmentioning

confidence: 99%

Rupture length and duration of the 2004 Aceh‐Sumatra earthquake from the phases of the Earth's gravest free oscillations

Lambotte

Rivera

Hinderer

2006

Geophysical Research Letters

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[1] The Aceh-Sumatra 2004 earthquake strongly excited the Earth's free oscillations. Well separated split multiplets provide useful information on the earthquake source. Particularly, the phases of split singlets constrain the duration, rupture length and mean rupture velocity. We analyze the initial phases of some of the Earth's gravest free oscillations ( 0 S 2 , 0 S 3 , 0 S 0 and 1 S 0 ) in order to constrain the space-time finiteness of the source. We use recordings of vertical broadband seismometers and superconducting gravimeters from several worldwide geophysical networks. We estimate a rupture length of about 1220 km, a source time duration of about 500 s, and a mean rupture velocity of 2.4 km/s. Citation: Lambotte, S., L. Rivera, and Introduction[2] Soon after the great Chilean 1960 earthquake, Benioff et al. [1961], Ness et al. [1961] and Alsop et al. [1961] reported accurate measurements of spectral peaks that were associated with the Earth's gravest free oscillations. Recordings of this event also provided the first observational evidence for splitting of the lowest frequency modes. This splitting was attributed to the Earth's rotation by Backus and Gilbert [1961] and Pekeris et al. [1961]. Splitting effect due to ellipticity was introduced later on by Usami and Satô [1962] and Dahlen [1968].[3] The 2004 Aceh-Sumatra earthquake is the largest earthquake ever recorded since the 1960 Chilean earthquake and the 1964 Alaskan earthquake. Initial teleseismic bodywave studies suggested a rupture length of about 400-600 km (e.g., J. [Ni et al., 2005;Lomax, 2005] suggest a longer rupture of over 1200 -1300 km. This is consistent with the location of the aftershocks that extend more than 1000 km northward from the epicenter.[4] Free oscillations are widely used to study the Earth's structure [e.g., Ritzwoller et al., 1986;Woodhouse et al., 1986;Smith and Masters, 1989;Widmer et al., 1992]. Free oscillations with periods between 50 and 300 s are also used to study source parameters of moderate and large earthquakes [e.g., . The gravest normal modes can also provide information on the focal mechanism, seismic moment [Abe, 1970;Geller and Stein, 1977;Kedar et al., 1994;Stein and Okal, 2005], duration and length of the earthquake rupture. Phases of free oscillations are seldom explicitly studied, except for the radial modes, [see Park et al., 2005a]. We demonstrate in this study how it is possible to retrieve the spatio-temporal extension of the rupture using the initial phases of some of the gravest modes ( 0 S 2 , 0 S 3 , 0 S 0 and 1 S 0 ) excited by the Aceh-Sumatra 2004 earthquake. Basics[5] Deviations from spherical symmetry such as rotation, ellipticity and lateral heterogeneities remove the degeneracy and hence force free oscillations multiplets to split. In general, each singlet within a multiplet will have a slightly different central frequency. Generally, to model the split spectrum of a multiplet, it is necessary to account for the Earth's rotation, ellipticity and heterogeneity. For the gravest m...

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“…Backus & Gilbert (1961), MacDonald & Ness (1961, and Pekeris, Alterman & Jarosch (1961) introduced Earth's rotation into the problem; Dahlen (1968) studied the effect of ellipticity of figure on a rotating Earth; Dahlen (1969) then proceeded further by considering the problem of quasi-degeneracy of these two perturbations ;Zharkov & Lyubimov (1970a, b) and Madariaga (1971Madariaga ( , 1972 treated the first-order ordinary degenerate perturbation of lateral inhomogeneities on a spherically symmetric, non-rotating Earth; and Luh (1973a) gave a progress report (call it Report I hereafter) on quasi-degeneracy of a general lateral perturbation. This present study represents an extension of Report I by incorporating Earth's rotation into the problem and by presenting numerical results of first-order perturbation theory.…”

Section: Introductionmentioning

confidence: 99%

Normal Modes of a Rotating, Self-gravitating Inhomogeneous Earth

Luh

1974

Geophysical Journal International

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This present study focuses on three topics: (i) formulation of the firstorder perturbation problem of the Earth's normal modes; (ii) formulation of a variational method based on the solutions of first-order quasidegenerate perturbation calculation; and (iii) numerical results for both ordinary degenerate and quasi-degenerate multiplets using formulation (i). The Earth considered is a rotating, laterally-inhomogeneous Earth. To zeroth order, however, it is approximated by a spherically symmetric, non-rotating, elastic and isotropic sphere. Lateral perturbations and Earth's rotation are treated as perturbations. Examples of lateral perturbations are Earth's ellipticity of figure and lateral inhomogeneities. Because of the symmetry in the Earth model, unperturbed modes form multiplets with varying degrees of degeneracy. For almost all of these multiplets, the perturbation calculation need be carried out to first order. When one multiplet and another nearly coincide in their frequencies, first-order ordinary degenerate perturbation method must be replaced by first-order quasi-degenerate perturbation method. Since the latter is a more general form of the former, the first-order perturbation problem is completely solved. The perturbations considered in this study consist of Earth's rotation, ellipticity of figure, a continent-ocean discontinuity near the Earth's surface, and an artificially contrived discontinuity about the core-mantle interface. When the effect due to rotation becomes dominant, the first-order calculation becomes insufficient, and a variational method is posed. Such a method can be formulated from quasi-degenerate solutions. The numerical results show that ellipticity corrections are important and should always be included in the computations of the Earth's normal modes, that lateral inhomogeneities in the Earth's deep interior become an important perturbation for high-overtone multiplets, and that Coriolis coupling between a spheroidal and a toroidal multiplet overshadows all other types of coupling.( O 2 5 -Y ) ( U Y v) = 0. Note that w2 I and Y here differ from the ones in equation (2.3). There po and C were functions of three-dimensional space co-ordinates; here they are functions of

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Rotational Multiplets in the Spectrum of the Earth

Cited by 79 publications

References 6 publications

Acoustic spectra of a gas-filled rotating spheroid

Acoustic spectra of a gas-filled rotating spheroid

Rupture length and duration of the 2004 Aceh‐Sumatra earthquake from the phases of the Earth's gravest free oscillations

Normal Modes of a Rotating, Self-gravitating Inhomogeneous Earth

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