S. A. Kumakshev

A gravitational-tidal mechanism for the Earth’s polar oscillations

Perturbed, rotational-oscillational motions of the Earth induced by the gravitational torques exerted by the Sun and Moon are studied using a linear mechanical model for a viscoelastic rigid body. A tidal mechanism is identified for the excitation of polar oscillations, i.e., for oscillations of the angular-velocity vector specified in a fixed coordinate frame, attributed to the rotational-progressive motion of the barycenter of the Earth-Moon “binary planet” about the Sun. The main features of the oscillations remain stable and do not change considerably over time intervals significantly exceeding the precessional period of the Earth’s axis. A simple mathematical model containing two frequencies, namely, the Chandler and annual frequencies, is constructed using the methods of celestial mechanics. This model is adequate to the astrometric measurements performed by the International Earth Rotation Service (IERS). The parameters of the model are identified via least-squares fitting and a spectral analysis of the IERS data. Statistically valid interpolations of the data for time intervals covering from several months to 15–20 yr are obtained. High-accuracy forecasting of the polar motions for 0.5–1 yr and reasonably trustworthy forecasting for 1–3 yr demonstrated by observations over the last few years are presented for the first time. The results obtained are of theoretical interest for dynamical astronomy, geodynamics, and celestial mechanics, and are also important for astrometrical, navigational, and geophysical applications.

Forecasting the polar motions of the deformable Earth

A mathematical model for the complicated phenomenon of the polar oscillations of the deformable Earth that adequately describes the astrometric data of the International Earth Rotation Service is constructed using celestial mechanics and asymptotic techniques. This model enables us to describe the observed phenomena (free nutation, annual oscillations, and trends) simply and with statistical reliability. The model contains a small number of parameters determined via a least-squares solution using well-known basis functions. Interpolations of the polar trajectory for intervals of 6 and 12 yrs and forecasts for 1–3 yrs are obtained using the theoretical curve. The calculated coordinates demonstrate a higher accuracy than those known earlier.

Analysis of multifrequency effects in oscillations of the earth's pole

A least-squares analysis of measurements of the Earth-rotation parameters is used to interpolate these data in order to redict the polar motion using a basic mathematical model that includes two frequencies: the Chandler and annual frequencies. A model taking into account the oscillations induced by the influence of the Moon is considered. The manifestation of high-frequency lunar oscillations in the beat period is demonstrated, together with the feasibility of interpolating these oscillations over short time intervals. A comparative analysis of models taking into account the monthly and bi-weekly frequencies is presented. A reasonable model explaining anomalous phenomena in the six-year beating is proposed.

A Model for the polar motion of the deformable Earth adequate for astrometric data

Refined analytical expressions for the frequencies corresponding to the Chandler motion of the pole and the diurnal rotation of the deformable Earth are derived. Numerical estimates of the period and amplitude of the polar oscillations are presented. The trajectory of the Chandler polar motion derived via numerical modeling is in qualitative and quantitative agreement with experimental data from the International Earth Rotation Service (IERS). An evolutionary model describing slow variations in the Earth’s rotation parameters under the action of the dissipative moments of the tidal gravitational forces on time scales considerably longer than the precession period of the Earth’s axis is constructed. The axis of the Earth’s figure tends to approach the angular momentum vector of the proper rotation.

High-accuracy forecasting of the Earth’s polar motion

The fundamental astrometrical problem of high-accuracy interpolation of the trajectory of the Earth’s pole and construction of an adequate theoretical model for associated complex multifrequency oscillations are considered. Measurements of the Earth-rotation parameters demonstrate the possibility of adjusting the filtering algorithm to make it suitable for practical navigational applications associated with a need for reliable high-accuracy predictions over the required time scales (short-and medium-terms). Numerical simulations and tests of the procedure used to optimize the adjustment parameters are presented.

The main properties and peculiarities of the Earth’s motion relative to the center of mass

The methods of theoretical and celestial mechanics and mathematical statistics have been used to prove that the Earth’s motion relative to the center of mass, the polar wobble, in the principal approximation is a combination of two circumferences with a slow trend in the mean position corresponding to the annual and Chandler components. It has been established that the parameters (amplitude and phase shift) of the annual wobble are stable, while those of the Chandler component are less stable and undergo significant variations over the observed time intervals. It has been proven that the behavior of these polar motion parameters is attributable to the gravitational-tidal mechanisms of their excitation.

Implementation of the desired state of motion for a vessel with heavy liquid

A mode of precision control of a vessel with an ideal heavy liquid into the state of the desired motion is proposed and justified. The algorithm is implemented and tested for typical control functions. The application efficiency of the proposed approach for poorly controllable oscillation systems with an infinite number of degrees of freedom under conditions of resonance is established.

Mechanical model of the perturbed motion of the earth with respect to the barycenter

A mechanical model of pole oscillations that fits the current views and the International Earth Rotation Service (IERS) data is constructed. The gravitational tidal torques due to the Sun and the Earth allowing for viscoelastic deformation of the Earth are considered to be the main perturbing factors. A bestfit, sixpara� metric mathematical model that takes into account the trend, annual oscillations, and Chandler oscilla� tions is constructed based on the IERS data using the spectral and dispersion analysis of a fairly complicated oscillation process. Major oscillation characteristics are determined using massive calculations. The inter� polation algorithm has been tested over long time scales, in particular, from 2000 through 2012. Algo� rithms for longscale prediction of oscillations at intervals of 60-900 days are proposed. 1. The dynamical theory of the rotation of the Earth with respect to the barycenter underlies many astrometric studies (1-5). It is known from astronom� ical observations (since the second half of the 19th century or, possibly, far back in the past) that the Earths rotation axis changes its orientation over time with respect to both the body coordinate system and the inertial coordinate system. This implies that the poles and latitudes evolve within noticeable ranges over a year. Measurements indicate that components with strongly different frequency and amplitude character� istics are involved in the very complicated oscillation process. Thus, small oscillations of the angular veloc� ity vector in a specific Earthfixed coordinate system (the reference system) contain the major component with an amplitude that reaches 0.20

The flow of a viscous fluid inside a confuser with large opening angle

The formulation of the classical Jeffery–Hamel problem [1–3] on the radial steady flow of a viscous fluid in a plane confuser includes two dimensionless parameters—the opening angle and the Reynolds number. Traditionally, the case of small angles, which is of great importance in technological applications, was investigated by analytical and numerical methods. In the mathematical and natural-scientific aspects, it is interesting to solve the problem for any opening angle in a wide range of the Reynolds number. In this study, we report the new results of numerical-analytical investigation of viscous-fluid for arbitrary possible angles (including the “exotic regions” of a confuser) ranging from a half-space (a plane with an outflow aperture) to a whole space with two extremely close half-planes between which the fluid drains off. A number of the new qualitative features of the velocity and pressure profiles are established and discussed. 1. The investigation of the radial motion of a viscous incompressible fluid in a plane confuser with opening angle 2β (0 < β ≤ π) under the action of pressure applied at infinity reduces to solving the nonlinear boundary value problem with an additional condition of constant flow rate [4]:

Flow of a viscoplastic medium with low yield strength in a flat confuser

1. When the shear yield strength of a material tends to zero, the scalar constitutive relation of the viscoplastic medium (the Bingham–Il’yushin model) is reduced to a physically linear relation for viscous fluid. In this limit, the flow studied below, in a certain sense, tends to the classical Jeffery–Hamel flow. The steady motion of viscous fluid in a flat confuser and a diffuser have been studied intensively [1–6]. Sometimes, this motion was chosen as a support flow for the approximate solution of a more complicated problem, for example, the nonisothermal problem [3]. Perturbing the Jeffery– Hamel flow by a low yield strength can be classed as a problem of deformation stability against the perturbation of material functions [4, 5, 7].