On The Determination Of The Earth's Model – The Mean Equipotential Surface

The sea surface cannot be used as reference for Major Vertical Datum definition because its deviations from the ideal equipotential surface are very large compared to rms in the observed quantities. The quasigeoid is not quite suitable as the surface representing the most accurate Earth's model without some additional conditions, because it depends on the reference field. The normal Earth's model represented by the rotational level ellipsoid can be defined by the geocentric gravitational constant, the difference in the principal Earth's inertia moments, by the angular velocity of the Earth's rotation and by the semimajor axis or by the potential (U ₀ ) on the surface of the level ellipsoid. After determining the geopotential at the gauge stations defining Vertical Datums, gravity anomalies and heights should be transformed into the unique vertical system (Major Vertical Datum). This makes it possible to apply Brovar's (1995) idea of determining the reference ellipsoid by minimizing the integral, introduced by Riemann as the Dirichlet principle, to reach a minimum rms anomalous gravity field. Since the semimajor axis depends on tidal effects, potential U ₀ should be adopted as the fourth primary fundamental geodetic constant. The equipotential surface, the actual geopotential of which is equal to U ₀ , can be adopted as reference for realizing the Major Vertical Datum.     

Morning polar substorms and variations in the atmospheric electric field

The effects of morning magnetospheric substorms in the variations in near-Earth atmospheric electricity according to the observations of the electric field vertical component (E _z ), at Hornsund polar observatory (Spitsbergen). The E _z, data, obtained under the conditions of fair weather (i.e., in the absence of a strong wind, precipitation, and fog), are analyzed. An analysis of the observations indicated that the development of a magnetospheric substorm in the Earth’s morning sector is as a rule accompanied by positive deviations in E _z, independently of the Hornsund location: in the polar cap or at its boundary. In all considered events, Hornsund was located near the center of the morning convection vortex. In the evening sector, when Hornsund fell in the region of evening convection vortex, the development of a geomagnetic substorm was accompanied by negative deviations in E _z., It has been concluded that the variations in the atmospheric electric field E _z), at polar latitudes, observed during the development of magnetospheric substorms, result from the penetration of electric fields of polar ionospheric convection (which are intensified during a substorm) to the Earth’s surface.     

Some space gravity formulas and the dimensions and the mass of the Earth

Summary Adopting thePizzetti-Somigliana method and using elliptic integrals we have obtained closed formulas for the space gravity field in which one of the equipotential surfaces is a triaxial ellipsoid. The same formulas are also obtained in first approximation of the equatorial flattening avoiding the use of the elliptic integrals. Using data from satellites and Earth gravity data the gravitational and geometric bulge of the Earth's equator are computed. On the basis of these results and on the basis of recent gravity data taken around the equator between the longitudes 50° to 100° E, 155° to 180° E, and 145° to 180° W, we question the advantage of using a triaxial gravity formula and a triaxial ellipsoid in geodesy. Closed formulas for the space field in which a biaxial ellipsoid is an equipotential surface are also derived in polar coordinates and its parameters are specialized to give the international gravity formula values on the international ellipsoid. The possibility to compute the Earth's dimensions from the present Earth gravity data is the discussed and the value ofMG=(3.98603×10²⁰ cm³ sec^–2) (M mass of the Earth,G gravitational constant) is computed. The agreement of this value with others computed from the mean distance Earth-Moon is discussed. The Legendre polinomials series expansion of the gravitational potential is also added. In this series the coefficients of the polinomials are closed formulas in terms of the flattening andMG.Publication Number 327, and Istituto di Geodesia e Geofisica of Università di Trieste.     

Mean Earth'S Equipotential Surface From Topex/Poseidon Altimetry

The geopotential value of W ₀ = (62 636 855.611 ± 0.008) m ² s ^–2 which specifies the equipotential surface fitting the mean ocean surface best, was obtained from four years (1993 - 1996) of TOPEX/POSEIDON altimeter data (AVISO, 1995). The altimeter calibration error limits the actual accuracy of W ₀ to about (0.2 - 0.5) m ² s ^–2 (2 - 5) cm. The same accuracy limits also apply to the corresponding semimajor axis of the mean Earth's level ellipsoid a = 6 378 136.72 m (mean tide system), a = 6 378 136.62 m (zero tide system), a = 6 378 136.59 m (tide-free). The variations in the yearly mean values of the geopotential did not exceed ±0.025 m ² s ^–2 (±2.5 mm).     

Semidiurnal and Long-Period Components in Polar Motion Connected with the Triaxiality of Ellipsoids of Inertia of the Earth and Planets

The influence of the Earth's inertia ellipsoid triaxiality, both on the motion of the Angular Velocity Vector with respect to Tisserand's reference frame (hereinafter – AVV,T), the motion of the same vector in Space (hereinafter – AVV,S) as well as on the motion of Tisserand's Reference Frame in Space (hereinafter – TRF,S) is considered. It is shown that the observable effects are as follows: (1) in all three types of motions (AVV,T), (AVV,S) and (TRF,S) there are significant (observable) semidiurnal components which may be detected by modern GPS- and VLBI-measurements. Additionally, there is a long-periodic perturbation of (TRF,S) motion which is rather small for the Earth, but may be significant for the other planets of the Solar System (if the triaxiality of the inertia ellipsoid is sufficiently large and whenever the orbital and diurnal rotations are synchronous and, at the points of equinoxes, the orientation of the planet's principal equatorial moments of inertia is invariable in space). For the case of the Earth, the results of our calculations are compared with the results of modern GPS-observations.     

Nonlinear plasma wave processes observed in the polar cusp

We present observations of electric and magnetic field variations from proton (about few Hz) to electron cyclotron frequencies (about few kHz) obtained by STAFF instrument on Cluster satellites during two cusp crossings, at ∼6 R _E altitude, in September 2002. The cusp was identified by the presence of intensive fluxes of counter streaming electrons with low energies and broadband wave activity which is typical for this region. Special attention is given for the interval of measurements when the waveform of the magnetic field fluctuations was taken in this region by CLUSTER satellites. The wave has been processed using the wavelet and bispectral analysis. Results showing the cascade of turbulence and wave-wave interactions are presented in this paper. A three wave process can be responsible for the broadening of the wave spectra in the polar cusp.     

Gravity potential at the Major Vertical Datum as primary geodetic constant

Summary On the basis of the fundamental relations of the Molodensky's Earth's figure theory (1945), admitting the inequality of the gravity potentials at the Major Vertical Datum W₀ and on the surface of the reference level ellipsoid U₀, and taken into account that potential W₀ enters into equations directly, it is recomended, W₀ should be adopted as a primary geodetic constant. Parameters of the best fitting ellipsoid are not needed for the solution of geodetic problems and for the investigation of the Earth's gravity field. The reason for requiring the reference and actual fields be close is only that the boundary-value problem can be solved in the linear approximation. Dedicated to the Memory of M.S. Molodensky Contribution to the I.A.G. Special Commission SC3 Fundamental Constants (SCFC).     

The Polar Cap (PC) indices: Relations to solar wind parameters and global magnetic activity

The polar geomagnetic activity resulting from solar wind–magnetosphere interactions can be characterized the Polar Cap (PC) indices, PCN and PCS. PC index values are derived from polar magnetic variations calibrated on a statistical basis such that the index approximate values in units of mV/m of the interplanetary “geo-effective” (or “merging”) electric field (E_M) conveyed by the solar wind. The timing and amplitude relations of the PC index to solar wind plasma and magnetic field parameters are reported. The solar wind effects are parameterized in terms of the geo-effective electric field (E_M) and the dynamical pressure (P_DYN). The PC index has a delayed and damped response to E_M variations and display saturation-like effects for E_M values exceeding 10 mV/m. Steady or slowly varying levels of solar wind dynamical pressure have little or no impact on the PC index above the effects related to E_M for which the solar wind velocity is also a factor. Sharp increases in the dynamical pressure generate impulsive variations in the PC index comprising a initial negative impulse of 5–10 min duration followed by a positive impulse lasting 10–20 min. Typical amplitudes of both the negative and the positive impulses are 0.2–0.5 units. A sharp decrease in the pressure produces the inverse sequence of pulses in the PC index. Auroral substorm activity represented by the AL index level has a marked influence on the average PC/E_M level at the transition from very quiet (AL0 nT) to disturbed conditions while more or less disturbed conditions (AL<−100 nT) have no systematic effect on the average PC/E_M values. At distinct substorm events the PC/E_M ratio has a minimum (0.8) in the pre-onset phase at around 20 min before substorm onset. The average ratio gradually increases in the expansion phase to reach a maximum value (1.1) at around 40 min after substorm onset (or 20 min after the largest (negative) peak in AL). At substorm recovery during the next 2 h the PC/E_M ratio decreases. Finally, we report on the application of polar magnetic variations to model the disturbance storm time (Dst) index development during magnetic storms by using the PC index as a source function to quantify the energy input to the ring current representing accumulated storm energy and characterized by the Dst index.     

On the physical meaning of the zonal components of the geopotential

The MacCullagh equation is of key importance in the study of the Earth. It connects the geometrical and the physical properties of the Earth through the geodynamical shape factor J₂. This second zonal geopotential coefficient is closely related to the flattening and to the angular spin velocity of the Earth as well as to its equatorial (A) and polar (C) moments of inertia. Through these moments of inertia the gravitational potential V is connected to the mass density distribution within the Earth.The main target of the present study is to obtain a generalized form of the MacCullagh equation for even orders n ≥ 2 by including the higher order zonal coefficients J_n connected with the higher (n ≥ 2) degree moments of inertia C_n and A_n. The higher the degree n, the higher is the weight of the near-surface (i.e. shallow) mass density distribution in J_n. The second part of this contribution deals with the temporal variations of J_n, dJ_n/dt for n ≥ 2.     

Oscillations with planetary wave periods in variations in the ionospheric parameters over Eastern Siberia

The results of a periodogramanalysis of the variations in the ionospheric parameters, measured using the vertical radio sounding method at midlatitude Irkutsk observatory (Eastern Siberia), are presented. The 1984–1986 period of observations was used. It has been indicated that the statistically significant oscillations with periods typical of planetary waves are present in the variations in f ₀E_s, f _bE_s, h′E_s, f _min, f ₀F2, and h′F.

     

Temporal variations of rotational origin in the absolute value of gravity

Summary When correcting precise gravity measurements for polar motion, the Earth's rotational deformation must be considered, as this will increase the correction based on a rigid Earth by about 15%. Conversely the gravity observations can be used to estimate the Love numbers h₂ and k₂ at the Chandler frequency.     

The zonal E×B plasma drift effects on the low latitude ionosphere electron density at solar minimum near equinox

The F2-layer peak density, NmF2, and peak altitude, hmF2, which were observed by 12 ionospheric sounders during the 20 September 1964 geomagnetically quiet time period at solar minimum are compared with those calculated by the three-dimensional time-dependent theoretical model of the Earth's low and middle latitude ionosphere and plasmasphere. The modeled NmF2 are also compared with those measured during the geomagnetically quiet time periods of 12–15, 18–21, and 26 September 1964 to take into account observed day-to-day ionospheric variability. Major features of the data are reproduced by the model if the corrected HWM90 neutral wind is used. The changes in NmF2 due to the zonal E×B plasma drift are found to be less than 20% in the daytime low latitude ionosphere. The model, which does not take into account the zonal E×B plasma drift, underestimates night-time NmF2 up to the maximum factor of 2 at low geomagnetic latitudes. The night-time increase of NmF2 caused by the zonal E×B plasma drift is less pronounced at −20° and 20° geomagnetic latitudes in comparison with that between −10° and 10° geomagnetic latitude. The longitude dependence of the calculated night-time low latitude influence of the zonal E×B plasma drift on NmF2 is explained in terms of the longitudinal asymmetry in B (the eccentric magnetic dipole is displaced from the Earth's center and the Earth's eccentric tilted magnetic dipole moment is inclined with respect to the Earth's rotational axis), and the variations of the wind induced plasma drift and the meridional E×B plasma drift in geomagnetic longitude. The difference between the hmF2 values calculated by including the effect of zonal E×B drift and that obtained when it is excluded does not exceed 19 km in the low latitude ionosphere. Over the geomagnetic equator the zonal E×B plasma drift produces the maximum increase in the electron density by a factor of 1.06–1.48 and 1.05–1.30 at 700 and 1000 km altitude, respectively, and this increase is not significant above about 1500 km. Changes in the vertical electron content, VEC, caused by the zonal E×B plasma do not exceed 16% during the day, while the value of the night-time VEC is increased up to a factor of 1.4 due to this drift. The maximum effects of the zonal E×B plasma drift on the night-time electron density derived from the model results corresponding to solar minimum and maximum are quite comparable.     

Ozone content of the atmosphere. 1. Climatological characteristics of the total ozone content in the 100–110° E longitudinal zone

The results of a periodogramanalysis of the variations in the ionospheric parameters, measured using the vertical radio sounding method at midlatitude Irkutsk observatory (Eastern Siberia), are presented. The 1984–1986 period of observations was used. It has been indicated that the statistically significant oscillations with periods typical of planetary waves are present in the variations in f ₀E_s, f _bE_s, h′E_s, f _min, f ₀F2, and h′F.     

Crop evapotranspiration in the Nile Delta under different irrigation methods

Eddy correlation measurements within the Nile Delta allowed the determination of evapotranspiration (E) for seven crops (rice, maize, cotton, sugar beet, berseem, wheat and fava beans) using basin irrigation (BI), furrow irrigation (FI), BI with increased intervals (BI_i), FI with increased intervals (FI_i), strip irrigation (SI) and drip irrigation (DI). Total E values over the cropping season for rice (BI, BI_i) were the highest (>600 mm), while those for sugar beet (DI), maize (SI and DI) and berseem (BI_i) were the lowest (<250 mm). The differences were due to a combination of atmospheric demand, soil moisture, the presence of surface standing water, root depth, and the length and timing of the cropping season. The DI and SI methods had the advantage for water saving, while the FI_i and BI_i methods were effective for crops with shallow root lengths. Estimated annual E was 566–828 mm/year (water-saving irrigation) and 875–1225 mm/year (conventional irrigation).     

Inter‐annual variations and factors controlling evapotranspiration in a temperate Japanese cypress forest

This study was undertaken to evaluate the effects of climatic variability on inter‐annual variations in each component of evapotranspiration (ET) and the total ET in a temperate coniferous forest in Japan. We conducted eddy covariance flux and meteorological measurements for 7 years and parameterized a one‐dimensional multi‐layer biosphere‐atmosphere model (Kosugi et al., 2006 ) that partitions ET to transpiration (Tr), wet‐canopy evaporation (E_wet), and soil evaporation (E_soil). The model was validated with the observed flux data. Using the model, the components of ET were estimated for the 7 years. Annual precipitation, ET, Tr, E_wet, and E_soil over the 7 years were 1536 ± 334 mm, 752 ± 29 mm, 425 ± 37 mm, 219 ± 34 mm, and 108 ± 10 mm, respectively. The maximum inter‐annual fluctuation of observed ET was 64 mm with a coefficient of variance (CV) of 2.7%, in contrast to relatively large year‐to‐year variations in annual rainfall (CV = 20.1%). Tr was related to the vapour pressure deficit, incoming radiation, and air temperature with relatively small inter‐annual variations (CV = 8.2%). E_soil (CV = 8.6%) was related mainly to the vapour pressure deficit. E_wet was related to precipitation with large inter‐annual variations (CV = 14.3%) because of the variability in precipitation. The variations in E_wet were counterbalanced by the variations in Tr and E_soil, producing the small inter‐annual variations in total ET. Copyright © 2016 John Wiley & Sons, Ltd.     

Evolution of non-linear α 2-dynamos and taylor's constraint

A key non-linear mechanism in a strong-field geodynamo is that a finite amplitude magnetic field drives a flow through the Lorentz force in the momentum equation and this flow feeds back on the field-generation process in the magnetic induction equation, equilibrating the field. We make use of a simpler non-linear?α?²-dynamo to investigate this mechanism in a rapidly rotating fluid spherical shell. Neglecting inertia, we use a pseudo-spectral time-stepping procedure to solve the induction equation and the momentum equation with no-slip velocity boundary conditions for a finitely conducting inner core and an insulating mantle. We present calculations for Ekman numbers (E) in the range 2.5× 10^?3 to 5.0× 10^?5, for?α?=α ₀cos?θ?sin?π?(r?r_i ) (which vanishes on both inner and outer boundaries). Solutions are steady except at lower E and higher values of?α?₀. Then they are periodic with a reversing field and a characteristic rapid increase then equally rapid decrease in magnetic energy. We have investigated the mechanism for this and shown the influence of Taylor's constraint. We comment on the application of our findings to numerical hydrodynamic dynamos.     

Computation of Ray Amplitudes in Inhomogeneous Media with Curved

The TOPEX/POSEIDON (T/P) satellite altimeter data from January 1, 1993 to January 3, 2001 (cycles 11–305) was used for investigating the long-term variations of the geoidal geopotential W ₀ and the geopotential scale factor R ₀ = GM÷W ₀ (GM is the adopted geocentric gravitational constant). The mean values over the whole period covered are W ₀ = (62 636 856.161 ± 0.002) m²s^-2, R ₀ = (6 363 672.5448 ± 0.0002) m. The actual accuracy is limited by the altimeter calibration error (2–3 cm) and it is conservatively estimated to be about ± 0.5 m²s^-2 (± 5 cm). The differences between the yearly mean sea surface (MSS) levels came out as follows: 1993–1994: –(1.2 ± 0.7) mm, 1994–1995: (0.5 ± 0.7) mm, 1995–1996: (0.5 ± 0.7) mm, 1996–1997: (0.1 ± 0.7) mm, 1997–1998: –(0.5 ± 0.7) mm, 1998–1999: (0.0 ± 0.7) mm and 1999–2000: (0.6 ± 0.7) mm. The corresponding rate of change in the MSS level (or R ₀) during the whole period of 1993–2000 is (0.02 ± 0.07) mm÷y. The value W ₀ was found to be quite stable, it depends only on the adopted GM, and the volume enclosed by surface W = W ₀. W ₀ can also uniquely define the reference (geoidal) surface that is required for a number of applications, including World Height System and General Relativity in precise time keeping and time definitions, that is why W ₀ is considered to be suitable for adoption as a primary astrogeodetic parameter. Furthermore, W ₀ provides a scale parameter for the Earth that is independent of the tidal reference system. After adopting a value for W ₀, the semi-major axis a of the Earth's general ellipsoid can easily be derived. However, an a priori condition should be posed first. Two conditions have been examined, namely an ellipsoid with the corresponding geopotential which fits best W ₀ in the least squares sense and an ellipsoid which has the global geopotential average equal to W ₀. It is demonstrated that both a-values are practically equal to the value obtained by the Pizzetti's theory of the level ellipsoid: a = (6 378 136.7 ± 0.05) m.     

Diurnal variations of the temperature in the middle ionosphere

Summary For the temperature of the middle ionosphere (100–180 km) variations proportional to the solar height are supposed with a certain inertia. The coefficients of these variations are computed on the basis of some rocket experiments and the asymetry in the daily variations of the electron density in theE-layer, the intermediateE-F region and theF1-layer. The diurnal vairations of the temperature are obtained for different constant heights.     

Parameters of the Earth's tri-axial level ellipsoid

Summary Four parameters defining the Earth's tri-axial ellipsoid (E) have been derived on the basis of the condition that the gravity potential on E be constant and equal to the actual geopotential value (W₀) on the geoid. The geocentric gravitational constant, the angular velocity of the Earth's rotation, the actual 2nd degree geopotential Stokes parameters and W₀ are taken to be the primary geodetic constants defining E and its (normal) gravity field.     

To Pizzetti's Theory of the Level Ellipsoid

The topic of the Earth's reference body, which has now been established as Pizzetti's level rotational ellipsoid, is analysed. Such a body is fully determined by four parameters: a, GM, J ₂ and . At present, the largest discrepancy in determining these parameters occurs in the value of a, which may in future be replaced by the gravity potential of the mean sea level W _o, with respect to Brovar's condition.Pizzetti's four parameters of the reference body are determined by solving the Dirichlet boundary value problem. The Dirichlet problem has only a unique solution, which, however, can be expressed in infinitely many ways. It turns out that the most important part in the form of the solution is played by Lamé's conditions, which determine the type of ellipsoidal coordinates.The solutions given by Pizzetti, Molodensky and another variant are considered. The last variant leads to a simple formula for the potential of the reference ellipsoid, but the formulae for Lamé's coefficients are inconvenient. Of course, all the methods lead to identical solutions, but some of them are more convenient for the historical use of logarithms, whereas others are more appropriate for use in computers.