On conversion between potential and kinetic energy in the atmosphere

Summary The basic concept of synoptic statistical methods for construction of prognostic charts was outlined by the author in a previous paper. As a result of these investigations it was found that a high correlation exists between time and space means of contourheights of an isobaric surface (850 mb surface). As it has been shown later byPichler this result may be interpreted by assuming that the geopotential fields obeys a numerical solution of the second order homogenous differential equation for wave propagation (hyperbolic equation) provided the phase velocity is given by . SinceReuter has used for s=666 km and for t=24 hours the conclusion may be drawn that the phase velocity of the wave propagation has an order of magnitude of 5 m/sec. Actually for long waves in the westerlies such a value can be found on an average. The same method can be used for extended forecast procedures if the wave equation is set down for 5 days mean values. Theoretical considerations lead then to a prognostic formula for a 5 days mean chart (8a). This formula can be applied for a sufficient number of grid points in order to construct prognostic charts. The underlying assumption, namely that the mean geopotential field satisfies also a solution of the wave equation turns out to be quite accurate even if only average values of the phase velocity were used for the computation. The usefullness of the method is illustrated for two cases.

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Vortrag gehalten am 7. April 1961 auf der 9. Allgemeinversammlung der «Società Italiana di Geofisica e Meteorologia» (Genova, 6.8. April 1961).     

Atmosphere and Earth's rotation

The theoretical aspects of the transfer of angular momentum between atmosphere and Earth are treated with particular emphasis on analytical solutions. This is made possible by the consequent usage of spherical harmonics of low degree and by the development of large-scale atmospheric dynamics in terms of orthogonal wave modes as solutions of Laplace's tidal equations.An outline of the theory of atmospheric ultralong planetary waves is given leading to analytical expressions for the meridional and height structure of such waves. The properties of the atmospheric boundary layer, where the exchange of atmospheric angular momentum with the solid Earth takes place, are briefly reviewed. The characteristic coupling time is the Ekman spin-down time of about one week.The axial component of the atmospheric angular momentum (AAM), consisting of a pressure loading component and a zonal wind component, can be described by only two spherical functions of latitude : the zonal harmonicP ₂ ⁰ (), responsible for pressure loading, and the spherical functionP ₁ ¹ () simulating supperrotation of the zonal wind. All other wind and pressure components merely redistributeAAM internally such that their contributions toAAM disappear if averaged over the globe. It is shown that both spherical harmonics belong to the meridional structure functions of the gravest symmetric Rossby-Haurwitz wave (0, –1)^*. This wave describes retrograde rotation of the atmosphere within the tropics (the tropical easterlies), while the gravest symmetric external wave mode (0, –2) is responsible for the westerlies at midlatitudes. Applying appropriate lower boundary conditions and assuming that secular angular momentum exchange between solid Earth and atmosphere disappears, the sum of both waves leads to an analytical solution of the zonal mean flow which roughly simulates the observed zonal wind structure as a function of latitude and height. This formalism is used as a basis for a quantitative discussion of the seasonal variations of theAAM within the troposphere and middle atmosphere.Atmospheric excitation of polar motion is due to pressure loading configurations, which contain the antisymmetric functionP ₂ ¹ () exp(i) of zonal wavenumberm=1, while the winds must have a superrotation component in a coordinate system with the polar axis within the equator. The Rossby-Haurwitz wave (1, –3)^* can simulate well the atmospheric excitation of the observed polar motion of all periods from the Chandler wobble down to normal modes with periods of about 10 days. Its superrotation component disappears so that only pressure loading contributes to polar motion.The solar gravitational semidiurnal tidal force acting on the thermally driven atmospheric solar semidiurnal tidal wave can accelerate the rotation rat of the Earth by about 0.2 ms per century. It is speculated that the viscous-like friction of the geomagnetic field at the boundary between magnetosphere and solar wind may be responsible for the westward drift of the dipole component of the internal geomagnetic field. Electromagnetic or mechanical coupling between outer core and mantle may then contribute to a decrease of the Earth's rotation rate.     

Propagation of Rayleigh waves in an axially symmetric heterogeneous layer lying between two homogeneous halfspaces

Summary The propagation of Rayleigh type waves in an axially symmetric inhomogeneous layer lying between two halfspaces is studied. The halfspaces are supposed to be identical in their elastic properties. The variation of the parameters in the layer is assumed to be of the form where is a constant andz is the distance measured from one interface into the layer. With this assumption, the vector wave equation for the layer is separable. The solution is obtained in terms ofWhittaker's functions and the frequency equation of Rayleigh type waves is derived.     

Hemispheric simulation of the Asian summer monsoon

A three-level, -plane, filtered model is used to simulate the Northern Hemisphere summer monsoon. A time-averaged initial state, devoid of sub-planetary scale waves, is integrated through 30 days on a 5° latitude-longitude grid. Day 25 through day 30 integrations are then repeated on a 2.5° grid. The planetary-scale waves are forced by time-independent, spatially varying diabatic heating. Energy is extracted via internal and surface frictional processes. Orography is excluded to simplify synoptic-scale energy sources.During integration the model energy first increases, but stabilizes near day 10. Subsequent flow patterns closely resemble the hemisphere summer monsoon. Climatological features remain quasi-stationary. At 200 mb high pressure dominates the land area, large-scale troughs are found over the Atlantic and Pacific Oceans, the easterly jet forms south of Asia, and subtropical jets develop in the westerlies. At 800 mb subtropical highs dominate the oceans and the monsoon trough develops over the Asian land mass. The planetary scales at all levels develop a realistic cellular structure from the passage of transient synoptic-scale features, e.g., a baroclinic cyclone track develops near 55°N and westward propagating waves form in the easterlies.Barotropic redistribution of kinetic energy is examined over a low-latitude zonal strip using a Fourier wave-space. In contrast to higher latitudes where the zonal flow and both longer and shorter waves are fed by barotropic energy redistribution from the baroclinically unstable wavelengths, the low-latitude waves have a planetary-scale kinetic energy source. Wave numbers 1 and 2 maintain both the zonal flow and all shorter scales via barotropic transfers. Transient and standing wave processes are examined individually and in combination.Wave energy accumulates at wave numbers 7 and 8 at 200 mb and at wave number 11 in the lower troposphere. The 800-mb waves are thermally indirect and in the mean they give energy to the zonal flow. These characteristics agree with atmospheric observation. The energy source for these waves is the three wave barotropic transfer. The implications of examining barotropic processes in a Fourier wave-space, vice the more common approach of separating the flow into a mean plus a deviation are discussed.     

Variations in zonal mean and planetary wave properties of the stratosphere and links with the troposphere

Zonal mean data and amplitudes and phases of planetary zonal waves were derived from daily hemispheric maps for tropospheric and stratospheric levels, for the four winters 1975–76 to 1978–79. Important year-to-year fluctuation in zonal means and wave activity are described, most notable of which are the changes from 1975–76 to 1976–77. Comparison of the relative strengths of the stratospheric and tropospheric jet streams shows a strong negative correlation (–0.8) between monthly mean zonal stratospheric winds (at 10 mb, 65°N) and zonal tropospheric winds (at 200 mb, 32.5°N, in the jet core) and a positive correlation (+0.7) between the stratospheric 10 mb winds and the tropospheric 200 mb winds at 65°N. Parameters correlated were the departures from the climatological mean zonal winds. The structure of correlation between wave amplitudes in the same wave number (1, 2) at different altitudes and between wave numbers 1 and 2 is investigated. We find a high correlation (+0.93) between wave 1 in the stratosphere (10 mb height) and wave 2 (height) in the troposphere at 65°N; but only a weak correlation (+0.2) between wave 1 amplitudes in the stratosphere and troposphere. These results suggest the possible importance of wave-wave interactions in processes linking the stratosphere and troposphere. The wave correlations presented here are based on comparisons of monthly means of daily amplitudes; the correlation structure in individual wave developments may differ, in view of the likelihood of altitudinal lags in wave amplification.     

M m: Extension to Love waves of the concept of a variable-period mantle magnitude

We extend to Love waves the concept of the mantle magnitudeM _mintroduced recently for Rayleigh waves. Spectral amplitudesX() of Love waves in the 50–300 s period range are measured on broad-band records from major events. A distance correctionC _D, regionalized to reflect the influence of different tectonic paths, and a source correctionC _S, compensating for the variation of excitation with period are effected; the exact geometry and depth of the event are however ignored. The resulting expression

Ueber Herde elastischer Wellen in isotropen,homogenen Medien

Zusammenfassung 1) Es werden Multipollösungen der skalaren Wellengleichung ² f/t ² – c² div gradf=0 betrachtet. Einerseits kann man solche Lösungen direkt durch Kugelfunktionenn-ter Ordnung ausdrücken, anderseits aus der Einpollösungf=1/p F(t–p/c) durch Differentiation nachn Richtungen erhalten. Es wird der Zusammenhang zwischen den Ergebnissen der beiden Verfahren gezeigt. — 2) Für die Energiedichte und den Energiefluss durch Kugelflächen bei kleinen elastischen Verschiebungen werden Ausdrücke in Kugelkoordinaten angegeben. — 3) Für die Wellengleichung grad div –b ² rot rot werden rotationsfreie Multipollösungen angegeben und Ausdrücke für Energiedichte und Energiefluss hergeleitet. — 4) Das gleiche wird für divergenzfreie Multipollösungen durchgeführt. — 5) Es werden Multipole betrachtet, die weder rotationsfrei noch divergenzfrei sind. Als Spezialfälle werden Multipole mit zeitlich begrenzter und solche mit periodischer Erregung gezeigt, ferner Lösungen der Wellengleichung, die sowohl rotationsfrei wie divergenzfrei sind. — 6) Es wird gezeigt, wie man die elastischen Wellen, die im Sinne vonStokes von einem Herdgebiet endlicher Ausdehnung ausgehen, näherungsweise durch elastische Multipole darstellen kann. — 7) Es wird angedeutet, wie man durch Messung von Komponenten von oder u.s.w. in Punkten im Innern des Mediums die Erregung und Energie von elastischen Multipolen bestimmen kann. Ferner wird auf den Fall hingewiesen, wo ein rotationsfreier Einpol sich im Innern eines Halbraumes befindet und die Messungen an seiner Oberfläche ausgeführt werden.

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Summary (On foci of elastic waves in isotropic homogeneous media) — 1) Multiplets as solutions of the scalar wave equation ² f/t ² – c² div gradf=0 are considered. Such solutions can be obtained either directly by aid of spherical harmonics of ordern, or by differentiating the single polef=1/p F(t–p/c) with respect ton directions. The relations between the results of those two procedures are shown. — 2) In the case of small elastic displacements , the density of energy and the flow of energy through spherical surfaces are expressed by spherical coordinates. — 3) Multiplets which satisfy the equation of motion =a ² grad div b ² curl curl and the equation curl = 0 are given, and expressions for the density and flow of energy are found. — 4) The same is done with multiplets satisfying the equation of motion and the equation div = 0. — 5) General multiplets which satisfy the equation of motion are treated. As special cases, multiplets with excitation of finite length and multiplets with periodic excitation are considered, furthermore solutions of the equation of motion and of the equations curl = 0 and div = 0 are given. — 6) It is shown how elastic waves whose origin is a region of finite extension in the sense given byStokes, can be approximated by elastic multiplets. — 7) Some indications are given on the problem of how to find the functions of excitation and the energy of an elastic multiplet by measuring components of or etc., at points in the interior of the medium. The same problem is considered in the case of the single elastic pole. = grad 1/p F (t–p/a), if the measurements are made at the surface of an elastic half space.

     

A study of the kinetic energy budget of the troposphere over the Caribbean Sea and the Gulf of Mexico

Summary The budget of kinetic energy over the Caribbean Sea and the Gulf of Mexico during the winter (November–April) and Summer (May–October) half-years of 1960 is established for the finite atmospheric layers 1000/850, 850/700, 700/500, and 500/300 mb. Vertical transports, lateral outflow, and generation of kinetic energy are computed directly from twice-daily aerological soundings, while the dissipation term is obtained as a residual. The frictional dissipation in the boundary layer is independently computed from 1960 ship observations. Kinetic energy dissipation is of the order of 1·10³ ergs·cm^–2 in the layer 1000/850 mb, decreasing in the higher layers. The residual dissipation term would indicate a production of kinetic energy, particularly for the layer 500/300 mb in winter. This would appear consistent with results by other authors. Due to the uncertainties inherent in the computational procedures, however, only limited confidence can be given to the absolute values.

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Zusammenfassung Die Kinetische Energie-Bilanz über dem Amerikanischen Mittelmeer während der Winter-(November–April) und Sommerhälfte (Mai-Oktober) des Jahres 1960 wird für die Schichten 1000/350, 850/700, 700/500 und 500/300 mb untersucht. Vertikaltransporte, seitlicher Export und die Produktion von kinetischer Energie werden direkt von den zweimal täglichen Radiosondenaufstiegen berechnet, während die Vernichtung kinetischer Energie als Restglied der Energiegleichung bestimmt wird. Der Energieverbrauch in der Grenzschicht wird unabhängig auf Grund von Schiffsbeobachtungen des Jahres 1960 abgeschätzt. Die Vernichtung kinetischer Energie hat in der Schicht 1000/850 mb die Grössenordnung von 1·10³ ergs·cm^–2·sec^–1, und nimmt nach den höheren Schichten zu ab. Das Restglied der Energiegleichung zeigt eine Produktion kinetischer Energie vor allem für die Schicht 500/300 mb im Winter an. Das erscheint verträglich mit den Ergebnissen anderer Autoren. Wegen der in den Berechnungsverfahren liegenden Unsicherheiten kommt den Absolutwerten nur begrenzte Gültigkeit zu.

     

Internal gravity wave selection in the upper troposphere and lower stratosphere observed by the MU radar: Preliminary results

Marked wavelike variations of the lower stratospheric wind observed on 7–10 May, 1985 by an MST radar in Japan (by the MU radar) are analyzed assuming that they are induced by monochromatic internal inertio-gravity waves. These variations are mainly composed of two modes (periods: 22 and 24 hours), both of which have zonal phase velocities (C _X ) slower than the mean westerly wind (). A statistical analysis of the zonal phase velocity shows thatC _X above andC _X below the tropopause jet stream, which is considered to be a vivid proof of wave selection due to the tropospheric mean flow and upward wave emission from the tropopause jet. A comparison between the MU radar results and routine meteorological observations leads to the conclusion that the marked waves appear when the jet stream takes a maximum wind speed.     

Beitrag zur Frage einer solaren Steuerung geophysikalischer Vorgänge

Zusammenfassung Die starke Abhängigkeit der zeitlichen Folge erdmagnetischer Störungen und Erdbehen von einer Periode 34 ^d .19 wird an den Diagrammen der Figg. 1–5 veranschaulicht. Nach der harmonischen Analyse tritt die Periodenwelle in zwei bestimmten, um 180° gegeneinander versetzten Phasenlagen auf. Ein Vergleich der Amplituden mit derSchusterschen Expektanz lässt erkennen, dass ein Walten des Zufalls hierbei praktisch ausgeschlossen sein sollte. Die Periode ist identisch mit der vom Verfasser früher abgeleiteten Rotationsdauer 34 ^d .19 eines hypothetischen Sonnenkerns. Auch Perioden der Form treten auf ( _p =Umlaufszeit der einzelnen Planeten), wie hier nur an einem Beispiel für den Merkur (Fig. 7) gezeigt wird.

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Summary It is shown in the diagrams Figg. 1–5, that the temporal sequences of terrestrial magnetic storms and earthquakes are largely dependent from a period of 34.19 days. The harmonic analyse demonstrates, that there are two waves of this period with a difference of 180° between them. The comparison of the amplitudes of waves with the expectance as defined byA. Schuster shows, that the period should be a reality. This period is identical with the period of 34.19 days for the rotation of an hypothetical sun-core, discovered earlier by the author. It is illustrated only at the example of the planet Mercury (Fig. 7), that there exist also periods of the form: ( _p =period of the revolution of the single planet).

     

Wave to wave and wave to zonal mean flow kinetic energy exchanges during contrasting monsoon years

Fourier analysis of the monthly mean northern hemispheric geopotential heights for the levels 700 mb and 300 mb are undertaken for the months of April through to August. The wave to wave and wave to zonal mean flow kinetic energy interactions are computed for specified latitude bands of the northern hemisphere during the pre-monsoon period (April to May) and monsoon period (June through to August) for bad monsoon years (1972, 1974, 1979) and for years of good monsoon rainfall over India (1967, 1973, 1977). Planetary scale waves (waves 1 to 4) are the major kinetic energy source in the upper atmosphere during the monsoon months. Waves 1 and 2 in particular are a greater source of kinetic energy to other waves via both wave to wave interactions as well as wave to zonal mean flow interactions in good monsoon years than in bad monsoon years. The zonal mean flow shows significantly larger gains in the kinetic energy with a strengthening of zonal westerlies in good monsoon years than in bad monsoon years.     

Nonlinear Magnetoconvection and the Geostrophic Flow - Subcritical Solutions

The magnetoconvection problem under the magnetostrophic approximation is investigated as the nonlinear regime is entered. The model consists of a fluid filled sphere, internally heated, and rapidly rotating in the presence of a prescribed, axisymmetric, toroidal magnetic field. For simplicity only a dipole parity and a single azimuthal wavenumber (m = 2) is considered here. The leading order nonlinearity at small amplitude is the geostrophic flow U _g which is introduced to the previously linear model (Walker and Barenghi, 1997a, b). Walker and Barenghi (1997c) considered parameter space above critical and found that U _g acts as an equilibration mechanism for moderately supercritical solutions. However, for solutions well above critical a Taylor state is approached and the system can no longer equilibrate. More importantly though, in the context of this paper, is that subcritical solutions were found. Here subcritical solutions are considered in more detail. It was found that, at is strongly dependent on . ( is the critical value of the modified Rayleigh number is a measure of the maximum amplitude of the generated geostrophic flow while , the Elsasser number, defines the strength of the prescribed toroidal field.) R_m at proves to be the key measure in determining how far into the subcritical regime the system can advance.     

Seismic crustal study of the Oslo Rift

A seismic refraction investigation across the southern part of the Oslo Rift has been made, based on quarry blasts at three localities. The study shows a three-layered crust with the followingP-wave velocities: . the upper mantleP-wave celocity, is 8.07 km/s. The velocity-depth relationship for the uppermost crust, obtained by solving the Wiechert-Herglotz integral equation numerically, shows a continuously decreasing velocity gradient in the region of the Oslo Rift which approaches zero at a depth of 9 km, the corresponding increase in theP-wave velocity being from 5.55 km/s to 6.34 km/s. The interface separating the subsurface layer ( =6.60 km/s) from the uppermost layer , interpreted as the Conrad discontinuity, is essentially horizontal in the investigated part of the Oslo Rift at a depth of approximately 15 km. A deep crustal layer with aP-wave velocity of 7.10 km/s appears to be related to the rift, though the top of this layer extends somewhat eastwards beneath the Precambrian rocks from the southern part of the rift at a depth of approximately 20 km. The Moho discontinuity is elevated beneath the Oslo Region compared with the surrounding area. A broad regional gravity high of about 45 mgal is observed along the entire rift zone. It is suggested that this anomaly is caused by the elevation of the sub-Conrad and Moho discontinuities during the rifting processes.     

Fermat's Variational Principle for Anisotropic Inhomogeneous Media

Fermat's variational principle states that the signal propagates from point S to R along a curve which renders Fermat's functional (l) stationary. Fermat's functional (l) depends on curves l which connect points S and R, and represents the travel times from S to R along l. In seismology, it is mostly expressed by the integral (l) = (x ^k,x ^k ')du, taken along curve l, where (x ^k,x ^k ') is the relevant Lagrangian, x ^k are coordinates, u is a parameter used to specify the position of points along l, and x ^k ' = dx ^k÷du. If Lagrangian (x ^k,x ^k ') is a homogeneous function of the first degree in x ^k ', Fermat's principle is valid for arbitrary monotonic parameter u. We than speak of the first-degree Lagrangian ⁽¹⁾(x ^k,x ^k '). It is shown that the conventional Legendre transform cannot be applied to the first-degree Lagrangian ⁽¹⁾(x ^k,x ^k ') to derive the relevant Hamiltonian ⁽¹⁾(x ^k,p _k), and Hamiltonian ray equations. The reason is that the Hessian determinant of the transform vanishes identically for first-degree Lagrangians ⁽¹⁾(x ^k,x ^k '). The Lagrangians must be modified so that the Hessian determinant is different from zero. A modification to overcome this difficulty is proposed in this article, and is based on second-degree Lagrangians ⁽²⁾. Parameter u along the curves is taken to correspond to travel time , and the second-degree Lagrangian ⁽²⁾(x ^k, ^k ) is then introduced by the relation ⁽²⁾(x ^k, ^k ) = [⁽¹⁾(x ^k, ^k )]², with ^k = dx ^k÷d. The second-degree Lagrangian ⁽²⁾(x ^k, ^k ) yields the same Euler/Lagrange equations for rays as the first-degree Lagrangian ⁽¹⁾(x ^k, ^k ). The relevant Hessian determinant, however, does not vanish identically. Consequently, the Legendre transform can then be used to compute Hamiltonian ⁽²⁾(x ^k,p _k) from Lagrangian ⁽²⁾(x ^k, ^k ), and vice versa, and the Hamiltonian canonical equations can be derived from the Euler-Lagrange equations. Both ⁽²⁾(x ^k, ^k ) and ⁽²⁾(x ^k,p _k) can be expressed in terms of the wave propagation metric tensor g _ij(x ^k, ^k ), which depends not only on position x ^k, but also on the direction of vector ^k . It is defined in a Finsler space, in which the distance is measured by the travel time. It is shown that the standard form of the Hamiltonian, derived from the elastodynamic equation and representing the eikonal equation, which has been broadly used in the seismic ray method, corresponds to the second-degree Lagrangian ⁽²⁾(x ^k, ^k ), not to the first-degree Lagrangian ⁽¹⁾(x ^k, ^k ). It is also shown that relations ⁽²⁾(x ^k, ^k ) = ; and ⁽²⁾(x ^k,p _k) = are valid at any point of the ray and that they represent the group velocity surface and the slowness surface, respectively. All procedures and derived equations are valid for general anisotropic inhomogeneous media, and for general curvilinear coordinates x ⁱ. To make certain procedures and equations more transparent and objective, the simpler cases of isotropic and ellipsoidally anisotropic media are briefly discussed as special cases.     

Propagation of love waves in a non-homogeneous stratum of finite depth sandwitched between two semi-infinite isotropic media

Summary The aim of this paper is to study a problem in which the intermediate layer is non-homogeneous, the rigidity varying exponentially with depth i.e. ₂=₂ v ₀ ² e ^2pz , the density being constant, velocity varies also exponentially with depth according to the law =v ₀ e ^pz . The variability ofKH with the change of phase velocity is shown graphically.     

An algorithm for the optimum distribution of a regional seismic network — II. An analysis of the accuracy of location of local earthquakes depending on the number of seismic stations

Summary Seven optimal networks consisting of 4 to 10 stations are compared for a given region, where velocity-depth profiles and the distribution of seismic intensity are known. Assuming that the standard error of arrival time is _t =0.05 s and the standard errors of the parameters of velocity-depth profiles are equal to 5% of their values, the average standard errors of the origin time and focus coordinates are estimated. The application of optimum methods to the planning of seismic networks in the Lublin Coal Basin is presented, and maps of standard errors of origin time , depth and epicenter ( _xy ) for the case of an optimum network of 6 seismic stations are given.     

Estimate of earthquake hazard in the Vrancea (Romania) region

A maximum likelihood method is used to estimate the earthquake hazard parameters maximum magnitudeM _max, annual activity rate , and theb value of the Gutenberg-Richter equation in the Vrancea (Romania) region. The applied procedure permits the use of mixed catalogs with incomplete historical as well as complete instrumental parts, the consideration of variable detection thresholds, and the incorporation of earthquake magnitude uncertainty.Our imput data, comprises 105 historical earthquakes which occurred between 984 and 1934, and a complete data file containing 1067 earthquakes which occurred during the period 1935–30 August, 1986. The complete part was divided into four subcatalogs according to different thresholds of completeness. Only subcrustal events were considered, and dependent events were removed.The obtained value (=0.65) is at the lower range of the previously reported results, but it appears concurrent with conceptual and observational facts. The same concerns inferred value of _max = 7.8 and activity rate _4.0 = 5.34.     

Zur Interpretation von seismischen Refraktionsmessungen

Zusammenfassung Unter der Voraussetzung, dass die Frontgeschwindigkeitc eine stetige und monoton wachsende Funktion von der Tiefez ist, wird dargelegt, wie man aus einer gemessenen Laufzeitkurve () die zuc inverse Funktionz=z (c) auf einfache Weise berechnen kann. Weiter wird die Eindringtiefez _m in Funktion von ermittelt und abschliessend ein Beispiel gegeben.

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Summary Based on a recorded travel-time curve (), a simple direct method is developed for calculating the functionz=z (c), under the asumption that the wave velocityc is a regularly monotone increasing function of the depthz. Finally a rumerical example is given.

     

A model for the simulation of turbulent and eddy diffusion processes at heights of 0–65 km

A generalized turbulent diffusion model has been developed which evaluates the time rate of growth of a simulated cloud of particles released into a turbulent (i.e. diffusive) atmosphere. The general model, in the form of second-order differential equations, computes the three-dimensional size of the cloud as a function of time. Parameters which influence the cloud growth, and which are accounted for in the model equations, are: (1) length scales and velocity magnitudes of the diffusive field, (2) rate of viscous dissipation , (3) vertical stability as characterized by the relative adiabatic lapse rate (1/T)(g/C _p +T/z), and (4) vertical shear in the mean horizontal winds , and , for a given height and of spatial extent equal to that of the diffusing cloud. Sample results for near ground level and for upper stratospheric heights are given. For the atmospheric boundary layer case, the diffusive field is microscale turbulence. In the upper stratospheric case it is considered to be a field of highly interactive and dispersive gravity waves.     

On some problems of seismic crustal phase

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