Geomagnetic latitude aspects of the effect of the interplanetary parameters on geomagnetic storms

Summary In this work the previous author's results concerning the geomagnetic effect of the interplanetary parameters in dependence on geomagnetic latitude are verified, complemented and presented with better accuracy. Data of 7 intensive storms recorded in 1973–79 at 5 observatories with slight differences in local time and with the appropriate latitude distribution limited by real possibilities have been analysed. Even in these cases the derived values of the constants determining the dependence of storm-time variations of the geomagnetic field upon both the dynamic pressure of the solar wind(P) and the interplanetary electric field(E_y) vary relatively regularly with geomagnetic latitude. The anomaly of Dst and DR-variations from the Almeria Observatory (AE) evident in some intensive storms is pointed out here. Unlike the previous work the time characteristics () of the ring current decay have been studied from the standpoint of the main (_m) and recovery (_r) phases of the storm. This yields higher values of _r as compared to from the above mentioned work. On the other hand, a large decrease in the values of _r was observed in some cases at a latitude of about 40°, as in the earlier study. Actually this phenomenon does not occur in all intensive storms as could be expected. As to the investigated storms, _m seems to be independent of geomagnetic latitude and much lower in its magnitude than _r.     

Aspetto elettrogeosmotico bidimensionale variabile col tempo

Riassunto Fino ad oggi sono apparse solo teorie elettrogeosmotiche transitorie unidimensionali. Quì risolviamo un problema del genere bidimensionale, con elettrodi verticali cilindrici. Le curve rappresentative delle portate d'acqua catodicaq _K a cui perveniamo, denotano diminuzioni abbastanza rapide fino ai tempi dell' ordine dei «tempi elettrodici », (t=), portate che poi diminuiscono in modo molto lento. Si riscontrano quì più estesi «pianerottoli»q _K che non quelli delle monodimensionalità (t=/9). Inoltre, col crescere dei rapporti distanze eteropolari-raggi catodici, o col diminuire dei raggi catodici, tali portate (che si compongono dei contributi dei singoli anodi) s'incrementano nel modo diagrammato.

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Summary Till now only mono-dimensional transient electro-geoosmotical theories have been published. In this paper a solution is given for a problem of bi-dimensional type, with vertical cylindrical electrodes. The plots which represent the amounts of cathodic waterq _K show quite rapid lowerings as far as to reach time durations of the order of «electrodic times » (t=) volumes which, after, diminish very slowly. In this case more extendedq _K «lobbies» are found than those in mono-dimensional case (t=/9). Furthermore increasing the ratios heteropolar spacings—cathodic radii, or diminishing the cathodic radii, water discharges (which result from the contribution of the single anods) increase as plotted in the diagrams.

     

Analytical study of the energy rate balance equation for the magnetospheric storm-ring current

We present some results of the analytical integration of the energy rate balance equation, assuming that the input energy rate is proportional to the azimuthal interplanetary electric field, E_y, and can be described by simple rectangular or triangular functions, as approximations to the frequently observed shapes of E_y, especially during the passage of magnetic clouds. The input function is also parametrized by a reconnection-transfer efficiency factor (which is assumed to vary between 0.1 and 1). Our aim is to solve the balance equation and derive values for the decay parameter compatible with the observed Dst peak values. To facilitate the analytical integration we assume a constant value for through the main phase of the storm. The model is tested for two isolated and well-monitored intense storms. For these storms the analytical results are compared to those obtained by the numerical integration of the balance equation, based on the interplanetary data collected by the ISEE-3 satellite, with the values parametrized close to those obtained by the analytical study. From the best fit between this numerical integration and the observed Dst the most appropriate values of are then determined. Although we specifically focus on the main phase of the storms, this numerical integration has been also extended to the recovery phase by an independent adjust. The results of the best fit for the recovery phase show that the values of may differ drastically from those corresponding to the main phase. The values of the decay parameter for the main phase of each event, _m, are found to be very sensitive to the adopted efficiency factor, , decreasing as this factor increases. For the recovery phase, which is characterized by very low values of the power input, the response function becomes almost independent of the value of and the resulting values for the decay time parameter, _r, do not vary greatly as varies. As a consequence, the relative values of between the main and the recovery phase, _m/_r, can be greater or smaller than one as varies from 0.1 to 1.     

Turbulent “Dynamo” in the Passive Scalar Problem

A turbulent magnetic dynamo can be considered as the evolution of a vector field in a turbulent fluid flow. The problem of evolution of scalar fields (e.g., number density of small particles) in a turbulent fluid flow is similar to the turbulent magnetic dynamo. The dynamo instability results in generation of magnetic field. The most important effect which can cause a generation of mean magnetic field in a turbulent fluid flow is the -effect: = – (1/3) u · ( × u), where u is the turbulent velocity field with the correlation time . A similar instability in the passive scalar problem results in formation of large-scale inhomogeneous structures in a spatial distribution of particles due to the -effect: = u_p ( · u_p), where u _p is the random velocity field of the particles which they acquire in a turbulent fluid velocity field. The effect is caused by inertia of particles which results in divergent velocity field of the particles. This results in additional turbulent nondiffusive flux of particles. The mean-field dynamics of inertial particles are studied by considering the stability of the equilibrium solution of the derived evolution equation for the mean number density of the particles in the limit of large Péclet numbers. The resulting equation is reduced to an eigenvalue problem for a Schrödinger equation with a variable mass, and a modified Rayleigh-Ritz variational method is used to estimate the lowest eigenvalue (corresponding to the growth rate of the instability). This estimate is in good agreement with obtained numerical solution of the Schrödinger equation. Similar effects arise during turbulent transport of gaseous admixtures (or light noninertial particles) in a low-Mach-number compressible fluid flow. The discussed effects are important in planetary and atmospheric physics (cloud formation, pollutant dynamics, preferential concentration of particles in protoplanetary disks and also planetesimals in them).     

A note on the stress-dilatancy relation for simulated fault gouge

Theoretical constraints on the stress-dilation relation for a deforming Coulomb material requirev ifC=0 andv sin^-1( _m / _m ) always, wherev is the dilation angle, is the friction angle,C is cohesion, _m is the maximum shear stress, and _m is the mean effective stress. Recent laboratory measurements of friction and dilatancy of simulated fault gouge show that small amplitude shear-load cycling causes compaction and consolidation. Comparison of the data with theory indicates that such load cycling produces: (1) increased coefficient of friction (or friction angle), (2) increased cohesion, and (3) increased dilatancy rate (or dilation angle). Under certain conditions of load cycling without significant plastic shear strain accumulation ( ^p <0.005) we find thatv exceeds both and, in contrast to theory, sin^-1( _m / _m ). This result is interpreted in terms of enhanced cohesion and overconsolidation, which lead to residual stresses within the gouge. An analogy is drawn between these special loading conditions and those extant on natural faults. In particular, our results imply that jostling and minor stress variations associated with microearthquakes may produce strengthening of fault gouge and changes in the fault zone's stress-dilatancy relation. Hence, compaction associated with microseismicity may lead to subsequent dilation of fault gouge, even for faults with large displacement rates and large net offsets (e.g., San Andreas). In regions where such dilation persists over sufficient displacements (on the order of the critical slip distance for seismic faulting) it may tend to inhibit unstable slip.     

Source characteristics of the 1992 Nicaragua tsunami earthquake inferred from teleseismic body waves

We analyzed the broadband body waves of the 1992 Nicaragua earthquake to determine the nature of rupture. The rupture propagation was represented by the distribution of point sources with moment-rate functions at 9 grid points with uniform spacing of 20 km along the fault strike. The moment-rate functions were then parameterized, and the parameters were determined with the least squares method with some constraints. The centroid times of the individual moment-rate functions indicate slow and smooth rupture propagation at a velocity of 1.5 km/s toward NW and 1.0 km/s toward SE. Including a small initial break which precedes the main rupture by about 10 s, we obtained a total source duration of 110 s. The total seismic moment isM _o =3.4×10²⁰ Nm, which is consistent with the value determined from long-period surface waves,M _o =3.7×10²⁰ Nm. The average rise time of dislocation is determined to be 10 s. The major moment release occurred along a fault length of 160 km. With the assumption of a fault widthW=50 km, we obtained the dislocationD=1.3 m. From andD the dislocation velocity isD=D/0.1 m/s, significantly smaller than the typical value for ordinary earthquakes. The stress drop =1.1 MPa is also less than the typical value for subduction zone earthquakes by a factor of 2–3. On the other hand, the apparent stress defined by 2E _s /M _o , where andE _s are respectively the rigidity and the seismic wave energy, is 0.037 MPa, more than an order of magnitude smaller than . The Nicaragua tsunami earthquake is characterized by the following three properties: 1) slow rupture propagation; 2) smooth rupture; 3) slow dislocation motion.     

Coulomb constitutive laws for friction: Contrasts in frictional behavior for distributed and localized shear

We describe slip-rate dependent friction laws based on the Coulomb failure criteria. Frictional rate dependence is attributed to a rate dependence of cohesionc and friction angle . We show that differences in the stress states developed during sliding result in different Coulomb friction laws for distributed shear within a thick gouge layer versus localized shear within a narrow shear band or between bare rock surfaces. For shear within gouge, shear strength is given by =c cos + _n sin, whereas for shear between bare rock surfaces the shear strength is =c cos + _n tan, where and _n are shear and normal stress, respectively. In the context of rate-dependent Coulomb friction laws, these differences mean that for a given material and rate dependence of the Coulomb parameters, pervasive shear may exhibit velocity strengthening frictional behavior while localized shear exhibits velocity weakening behavior. We derive from experimental data the slip-rate dependence and evolution ofc and for distributed and localized shear. The data show a positive rate dependence for distributed shear and a negative rate dependence for localized shear, indicating that the rate dependence ofc and are not the same for distributed and localized shear, even after accounting for differences in stress state. Our analysis is consistent with the well-known association of instability with shear localization in simulated fault gouge and the observation that bare rock surfaces exhibit predominantly velocity weakening frictional behavior whereas simulated fault gouge exhibits velocity strengthening followed by a transition to velocity weakening with increasing displacement. Natural faults also exhibit displacement dependent frictional behavior and thus the results may prove useful in understanding the seismic evolution of faulting.     

Cyclic variations of the initial height of the sporadicE-layer at middle geographical latitudes

Summary The variations of the initialh E _s height are investigated in the solar cycle 1957–1968, deriving the regressive dependency:h E _s =121.4–6·10^–2 R referring to the median monthly values at a solar zenith angle =75°. The similar variations ofh E _s (R) during the day and night are interpreted as a domination of the sporadic layer formation from a redistribution of the day-time ionization and secondary participation of nightly ionizing sources. The analogous cyclich E _s andh E variations confirm this conclusion while the seasonal variations in the state of the sporadic layer show outlined dynamical effects. The comparatively not big cyclic variation in the spatial state of theE-region are considered to confirm the predominating ionizing action of the ultraviolet range (933–1038 Å) in the lower part of theE-region, while the soft X-radiation influences mainly the near maximum part of this region.     

The bayesian probabilistic prediction of the next earthquake in the ometepec segment of the mexican subduction zone

A predictive equation to estimate the next interoccurrence time () for the next earthquake (M6) in the Ometepec segment is presented, based on Bayes' theorem and the Gaussian process.Bayes' theorem is used to relate the Gaussian process to both a log-normal distribution of recurrence times () and a log-normal distribution of magnitudes (M) (Nishenko andBuland, 1987;Lomnitz, 1964). We constructed two new random variablesX=InM andY=In with normal marginal densities, and based on the Gaussian process model we assume that their joint density is normal. Using this information, we determine the Bayesian conditional probability. Finally, a predictive equation is derived, based on the criterion of maximization of the Bayesian conditional probability. The model forecasts the next interoccurrence time, conditional on the magnitude of the last event.Realistic estimates of future damaging earthquakes are based on relocated historical earthquakes. However, at the present time there is a controversy between Nishenko-Singh and Gonzalez-Ruiz-Mc-Nally concerning the rupturing process of the 1907 earthquake. We use our Bayesian analysis to examine and discuss this very important controversy. To clarify to the full significance of the analysis, we put forward the results using two catalogues: (1) The Ometepec catalogue without the 1907 earthquake (González-Ruíz-McNally), and (2) the Ometepec catalogue including the 1907 earthquake (Nishenko-Singh).The comparison of the prediction error reveals that in the Nishenko-Singh catalogue, the errors are considerably smaller than the average error for the González-Ruíz-McNally catalogue of relocated events.Finally, using the Nishenko-Singh catalogue which locates the 1907 event inside the Ometepec segment, we conclude that the next expected damaging earthquake (M6.0) will occur approximately within the next time interval =11.82 years from the last event (which occurred on July 2, 1984), or equivalently will probably occur in April, 1996.     

Verification and refinement of spectral magnitude calibrating functions for P-waves

Calibrating functions for determination of P-wave spectral magnitudes calculated by Duda and Yanovskaya (1994) on the basis of the IASP91 velocity model and the PREM Q-model are verified empirically. For this purpose IRIS broadband records for 120 earthquakes are used, each earthquake having been recorded at about 100 stations. The discrepancies indicate that anelastic absorption assumed in PREM is too high. New calibrating functions are calculated on the basis of the AK135 velocity and anelasticity models (Kennet et al., 1995; Montagner and Kennet, 1996), in which Q in the mantle is higher than in PREM. The verification of the new calibrating functions based on the same observations yields magnitude figures less depending on the epicentral distance. In addition the parameter ₂in the Liu-Anderson Q(T) model is estimated, proceeding from the assumption that on the average the radiated spectra comply with the ²-model. The value of ₂ was assumed to be 0.1 s in the analysis and its use resulted in the verification of the ²-model for the source spectrum.     

Flow of non-Newtonian fluid through circular tubes

Summary According to Newton's law of viscosity _y = Dv_y/dy. But experiments have shown that _y is indeed proportional to –dv _x/dy for all gases and for homogeneous nonpolymeric liquids. There are however, a few industrially important materials, e.g. plastics, asphalts, crystalline materials that are not described by the equation given by Newton's law of viscosity and they are referred to as non-Newtonian fluids. The steady state rheological behaviour of most fluids can be expressed by the generalised form, _y = –(dv_y/dy) where may be expressed as a function of eitherdv _x/dy or _y (where is independent of the rate of shear, the behaviour is Newtonian with =). Numerous empirical equations or models have been proposed to express the steady-state relation between _y anddv _x/dy. The flow of Newtonian fluids through circular tubes have been discussed before by many. Here we shall discuss the case of two such models of non-Newtonian fluids through circular tubes. The flow of fluids in circular tubes is encountered frequently in Physics, Chemistry, Biology and Engineering.     

Magnitude estimation of Koyna-Warna earthquakes, India

The paper presents the current state of magnitude estimation for Koyna earthquakes exceeding magnitude 3.0. We estimate coda duration magnitude from analogue seismograms recorded on the short period vertical (SPZ) seismometer at Hyderabad seismic observatory HYB and determine moment magnitude using very broad-band (VBB) data from the Geoscope station (HYB)and short period digital data from the local seismic network of NationalGeophysical Research Institute (NGRI) around the Koyna and Warna reservoirs.Firstly, the seismograms of 97 Koyna earthquakes exceeding magnitude 4.0 havebeen used to establish a new empirical coda duration magnitude scale which includes the higher order terms of log₁₀, where is the coda length in seconds. Four background noise levels (1, 2, 6 and 10 mm) areconsidered to estimate the coda duration. We found that the duration magnitudes for 1 mm background level are more stable than those for 2, 6 and 10 mm. The new coda duration magnitude (M_dnew) scale, for 1 mmlevel, is:M_dnew = –0.594 + 2.04 log₁₀ – 0.0435 (log₁₀)²The estimated M_dnew are compatible with the reported M_S values of the NGRI observatory and the m_b values of the United States Geological Survey (USGS). These magnitudes can be obtained within the standard deviation of ± 0.26 units of M_S (NGRI). A new relatively homogeneous catalog for Koyna earthquakes of M_dnew 4.0 is obtained. The momentmagnitudes for 34 Koyna-Warna events of M_dnew ranging from 3.0 to 5.4 have been estimated using two techniques. The first utilizes amplitudes of band-pass filtered (between 15 and 30 sec) velocity traces of moderate Koyna-Warna earthquakes of M_W} 4.4 to 5.4, we abbreviate the magnitude using M_A. The second is based on the S-wave spectrum of short period seismograms of local earthquakes (M_W < 3.8). Moment magnitudes estimated by spectral analysis mainly depend on the estimation of event's long-period spectral level and appears to saturate for moderate Koyna-Warnaearthquakes (M_W > 3.8). We recommend the use of both techniques whenever possible. The estimated moment magnitudes and M_dnew show an almost linear relationship with a standard deviation of ± 0.05.     

Electromagnetic scattering by a perfectly conducting half-plane in a conductive half-space

FollowingDmitriev (1960) a rigorous theoretical solution for the problem of scattering by a perfectly conducting inclined half-plane buried in a uniform conductive half-space has been obtained for plane wave excitation. The resultant integral equation for the Laplace transform of scattering current in the half-plane is solved numerically by the method of successive approximation. The scattered fields at the surface of the half-space are found by integrating the half-space Green's function over the transform of the scattering current.The effects of depth of burial and inclination, of the half-plane on the scattered fields are studied in detail. An increase in the depth of burial leads to attenuation of the fields. Inclination introduces asymmetry in the field profiles beside affecting its magnitude. Depth of exploration is greater for quadrature component. An interpretation scheme based on a phasor diagram is presented for the VLF-EM method of exploration for rich vein deposits in a conductive terrain.List of symbols x, y, z Space co-ordinates - Half-space conductivity - ₀ Free-space permeability - Excitation frequency (angular) - T Time - h Depth of the half-plane - a Inclination of the half-plane - E _x x-Directed total electric field - E _x ^p x-Directed primary electric field - E _xo ^p x-Directed primary electric field atz=0 directly over the half-plane - H _y y-Component of total magnetic field - H _y ^p y-Component of primary magnetic field - H _y0 ^p y-Component of primary magnetic field atz=0 directly over the half-plane - H _z z-Component of total magnetic field - H _z ^p z-Component of primary magnetic field - J _x Surface density ofx-directed scattering current - G Green's function - k ₀,K Wave numbers - u,u ₀,u ₁,u ₂ Functions - Space co-ordinate - s Variable in transform domain - Variable of integration - Normalized scattering current - Laplace transform of - N Normalized - , ₀, ₁, ₂ Functions - t Variable of integration - Skin depth - H Total magnetic field - H ^p Primary magnetic field - H ₀ ^p Primary magnetic field atz=0 directly over the half-plane - M,Q,R,S,U,V Functions - N ₁,N ₂ Functions     

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).

     

Solar activity dependence of ionospheric electron content and slab thickness using different solar indices

Ionospheric electron content (IEC) and slab thickness () data for the period 1977 to 1980 from Lunping (23.03°N; 121.90°E subionospheric) have been examined for their solar activity dependence. Local noontime monthly means as well as values for the 5 QQ days in a month have been examined separately with different solar indices, namely: solar EUV flux (170–190 Å),S _{10.7 cm} flux and sun spot number (SSN) on a seasonal basis. Both IEC and parameters exhibit better correlation with solar EUV andS _{10.7 cm} fluxes than with SSN for all seasons. IEC increases linearly with both EUV andS _{10.7 cm} flux whereas with SSN it shows a distinct nonlinear relationship during all seasons in both monthly mean and 5 QQ days' values. This study indicates that for correlating and predicting the variations (especially the medium term) in the ionospheric parameters, both EUV andS _{10.7 cm} fluxes have an advantage over SSN.     

A theoretical investigation of tropospheric ozone and stratospheric-tropospheric exchange processes

Summary Based on the global distribution of various surface types the mean tropospheric residence time of ozone is estimated as a function of latitude. Due to the land-sea distribution varies from 50 days in the northern hemisphere to 190 days in the southern hemisphere. For the stratospheric-tropospheric exchange a sinusoidal variation with season is assumed. The annual variation of tropospheric ozone thus gets a sine function from mean, amplitude and phase of whch the injection function for the particular latitude can be determined.     

A method of searching for radial velocity anomalies in the lower mantle using interfering signals

Radial velocity anomalies in the lower mantle that give rise to triplications in the travel-time curve for short-periodP waves will produce arrivals havingdT/d values that differe by roughly 0.2–0.5 s/deg. The first two arrivals associated with such triplications will be separated by less than one second over a distance range of 4°–10° they may not, therefore, be separable visually on single seismograms, so that their presence can only be inferred from some measurable property that depends on their mutual interference. If there are lateral variations in the regions of anomalous velocity gradients, the interfering signals will also have different azimuths of arrival. Using two synthetic wavelets we have investigated the effect of interference on bothdT/d and azimuth measurements at the Yellowknife Array. We found that if the interfering pulses have a dominant frequencyv, there is a range of time separations (0.30/v0.55/v) over which the measureddT/d and azimuth values may fluctuate by much more than the differences indT/d and azimuth between the interfering signals. We have evaluated the following empirically defined functions for three different primary signals, and for three different relative amplitudes of the interfering signals:f (t), the drift function, which expresses how the measured slownesses,p, and azimuths, , differ from the slownesses and azimuths of the primary wavelets; f(), the range function, which describes the behaviour of the upper and lower bounds ofp and as a function of the difference in arrival times of the signals, andf , studied the properties of these functions, and have outlined how these properties provide criteria based on the numerical and statistical characteristics of the arrival vectors, and on the waveform of the signal that will enable small radial velocity anomalies to be more clearly delineated.Contribution No. 863 from the Earth Physics Branch.     

Duration Magnitude Scale and Site Residuals for Northern Morocco

— The first empirical duration magnitude (M_D) formula is developed and tested for the Northern Morocco Seismic Network (NMSNET). This relationship is obtained by relating the IGN (Instituto Geografico National, Madrid) body-waves mbLg^IGN to the duration (), and the epicentral distance (), at 25 analogue stations of the NMSNET for 479 earthquakes with 2.5 mb 5.4, from March 1992 to February 2001. M_D estimates are significantly more precise while introducing a correction term for each of these stations, cSta_j. The magnitude for the i^th event (M_D)_i is the mean value of individual M_Dij=–0.14+1.63log₁₀(_ij)+0.031(_ij)+cSta_j. The cSta_j corrections reduce considerably the local site effects which influence the recorded durations and cause stations to either overestimate, or underestimate M_D up to 0.5 magnitude units. Average station M_D residuals (–cSta_j) are found to be independent of the distance from the epicenter to at least 10 degrees. It seems evident that regional geological features in the immediate behavior of stations have a systematic effect on the corresponding obtained residuals: older well-consolidated Precambrian crystalline rocks produce high negative residuals (shorter durations), younger unconsolidated sediments produce high positive residuals (longer durations), whereas, intermediate M_D site residuals appear to be the result of the effect of various factors, principally age and state of consolidation of the bedrock, combined with the local tectonic.     

Planar charged-particle trajectories in multipole magnetic fields

This paper provides a complete generalization of the classic result that the radius of curvature () of a charged-particle trajectory confined to the equatorial plane of a magnetic dipole is directly proportional to the cube of the particles equatorial distance () from the dipole (i.e. ³). Comparable results are derived for the radii of curvature of all possible planar chargedparticle trajectories in an individual static magnetic multipole of arbitrary order m and degree n. Such trajectories arise wherever there exists a plane (or planes) such that the multipole magnetic field is locally perpendicular to this plane (or planes), everywhere apart from possibly at a set of magnetic neutral lines. Therefore planar trajectories exist in the equatorial plane of an axisymmetric (m = 0), or zonal, magnetic multipole, provided n is odd: the radius of curvature varies directly as ⁿ⁼². This result reduces to the classic one in the case of a zonal magnetic dipole (n = 1). Planar trajectories exist in 2m meridional planes in the case of the general tesseral (0 < m < n) magnetic multipole. These meridional planes are defined by the 2m roots of the equation cos[m()–_n^m)] = 0, where _n^m = (1/m) arctan (h_n^m/g_n^m); g_n^m and h_n^m denote the spherical harmonic coefficients. Equatorial planar trajectories also exist if (n – m) is odd. The polar axis ( = O,) of a tesseral magnetic multipole is a magnetic neutral line if m > I. A further 2_m(n – m) neutral lines exist at the intersections of the 2_m meridional planes with the (n – m) cones defined by the (n – m) roots of the equation P_n^m(cos ) = 0 in the range 0 < 9 < , where P_n^m(cos ) denotes the associated Legendre function. If (n – m) is odd, one of these cones coincides with the equator and the magnetic field is then perpendicular to the equator everywhere apart from the 2m equatorial neutral lines. The radius of curvature of an equatorial trajectory is directly proportional to ⁿ⁼² and inversely proportional to cos[m(–)]. Since this last expression vanishes at the 2m equatorial neutral ines, the radius of curvature becomes infinitely large as the particle approaches any one of these neutral lines. The radius of curvature of a meridional trajectory is directly proportional to rⁿ⁺², where r denotes radial distance from the multiple, and inversely proportional to P_n^m(cos )/sin . Hence the radius of curvature becomes infinitely large if the particle approaches the polar magnetic neutral ine (m > 1) or any one of the 2m(n – m) neutral ines located at the intersections of the 2m meridional planes with the (n – m) cones. Illustrative particle trajectories, derived by stepwise numerical integration of the exact equations of particle motion, are pressented for low-degree (n 3) magnetic multipoles. These computed particle trajectories clearly demonstrate the non-adiabatic scattering of charged particles at magnetic neutral lines. Brief comments are made on the different regions of phase space defined by regular and irregular trajectories.Also Visiting Reader in Physics, University of Sussex, Falmer, Brighton, BN1 9QH, UK     

Generalized thermoelastic longitudinal waves in unbounded medium

Summary In this paper the generalized thermoelastic longitudinal waves and the temperature field set up due to coupling of the displacement and the temperature fields, with heat wave travelling with certain finite velocity, in an unbounded medium are studied. The thermoelastic displacement potential and the temperature field at any point are obtained in terms of the surface integrals involving the potential, the temperature and their normal derivatives.Notation x _i the cartesian coordinate system,i=1,2,3 - n (u _i) the displacement vector - (/x _i) the del operator - _t /t the derivative with respect to time - T _o the temperature corresponding to the natural stat of zero stress and strain - T Absolute temperature - c _e the specific heat - , Lamé's constants - ₀ the density - coefficient of linear thermal expansion - K thermal conductivity coefficient - _kk u - ₀ the relaxation time