Some hydrodynamic problems for a nepheloid zone

Summary A layer of a few hundred meters thickness with suspended matter (a nepheloid zone) was discovered byEwing andThorndike [4]³⁾ near the bottom on the continental slope of the North Atlantic. A downward pressure gradient is produced in this layer due to increment of water density with suspensoid. When only the Coriolis force balances with this pressure gradient, a bottom nepheloid current flows southwestward parallel to the depth contours with a velocity of about 10 (cm/sec) for a slope of one degree. The pressure gradient for fluid with locally variable density above a sloping bottom is treated and an extra term due to density gradient along the slope is derived. The vertical profiles of the nepheloid current with an effect on the vertical eddy viscosity are computed. Two kinds of vertical distributions of eddy viscosity are determined from the observed nepheloid distributions and used in the calculations: constant but different values at two layers and those increasing with height. The effect of the change of density along the bottom is treated by introducing dimensionless variables. Rossby number of the nepheloid current becomes about 10^–2 indicating inertia terms to be negligible. Rossby number of turbidity currents ranges from 2 (in a decaying area) to 5 (developing area), suggesting that inertia terms are more important than Coriolis terms. The trajectories of turbidity currents are computed from motion of a mass of mud under the Coriolis force and friction, and the results are applied to those inferred byHand andEmery [6] in the San Diego Through off California.LGO Contribution Number 925.     

Predicted near-field ground motion for dynamic stress-drop models

We propose a finite difference method, using a hexagonal grid, to compute displacements (stresses, velocities, accelerations) in the near-field of a 2-D in-plane stress-drop crack, in both whole space (constant stress-drop) and half-space (depth-dependent stress-drop). To exercise the method, the stress field distribution is evaluated for both fundamental 2-D shear cracks, anti-plane. In order to test the method's reliability, the results are compared with some analytical and numerical solutions available in the literature (Kostrov, 1964;Virieux andMadariaga, 1982). For the in-plane source, the results emphasize that the method can resolve the stress concentration due to the rupture front from the stress peak associated with the shear wave propagating in front of the crack. Synthetic motions are computed on the fault, but also in an infinite medium and at the free surface. The rather complex waveforms generated in the near-field, even by simple sources, emphasize the contribution of all wave terms (near, intermediate and far-field) to the motion. The presence of near-field and the numerical procedure explain the significant low frequency content of the computed seismograms. The set of treated problems proves the method is stable and accurate.     

Acceleration of mean zonal flows by planetary waves

The mechanism of acceleration of the mean zonal flow by a planetary wave is explained intuitively by considering the wave drag which a corrugated bottom feels when it excites the wave. The explanation is justified by solving the problem of vertical propagation of a planetary wave packet and the second order mean motion induced around it. The discussion is slightly extended to the case of small damping, to illustrate in a compact form the fact that the mean zonal acceleration is determined by a forcing due to wave transience plus that due to wave dissipation.The mean flow induced by a steady, dissipating planetary wave is discussed, and it is shown that it depends largely on the dissipation scale-height of the wave whether the northern region is heated or cooled. For example, if the wave velocity-amplitude increases upward in spite of dissipation, the induced easterly flow increases with height and the temperature of the northern region increases relative to that in the southern region. A similar point has been made byDunkerton (1979) in connection with westerly flows induced by Kelvin waves.The Lagrangian-mean motion induced by a planetary wave is briefly discussed in connection with the mechanism of acceleration of the mean zonal flow, in the case of a slowly varying wave packet. Further, in order el elucidate the effects of wave dissipation and time dependence of wave amplitude, the results obtained for a steady, dissipating wave and for a growing baroclinic wave are mentioned.     

The longitudinal proper motion of sunspots and the solar rotation rate

Summary The longitudinal proper motion of sunspot groups has been shown to be a function of their size and relative extension in longitude. Two theories are proposed to explain these effects, which theories assume that the wind circulation of the spots and the wind field surrounding the spots are intimately connected, and dynamically interact. The retardation in the motion of large spot groups is hypothesized to be due to the effect of the latitudinal variation of the Coriolis parameter (on a rotating spherical body) and an assumed vortical circulation around the spots. The more rapid longitudinal motion of extended spot groups is suggested to be the result of the preferential occurrence and development of such groups in regions of above-average horizontal wind and wind shear. If these theories are essentially correct, the real solar rotation rate—i.e., the longitude- and time-averaged fluid motion field—should be derived from the motions of small spot groups, and is 1% to 11/2% higher than the usually accepted rate whichNewton andNunn derived.     

On the Mathematical Analysis of an Elastic-gravitational Layered Earth Model for Magmatic Intrusion: The Stationary Case

In the early eighties Rundle (1980, 1981a,b, 1982) developed the techniques needed for calculations of displacements and gravity changes due to internal sources of strain in layered linear elastic-gravitational media. The approximation of the solution for the half space was obtained by using the propagator matrix technique. The Earth model considered is elastic-gravitational, composed of several homogeneous layers overlying a bottom half space. Two dislocation sources can be considered, representing magma intrusions and faults. In recent decades theoretical and computational extensions of that model have been developed by Rundle and co-workers (e.g., Fernández and Rundle, 1994a,b; Fernández et al., 1997, 2005a; Tiampo et al., 2004; Charco et al., 2006, 2007a,b). The source can be located at any depth in the media. In this work we prove that the perturbed equations representing the elastic-gravitational deformation problem, with the natural boundary and transmission conditions, leads to a well-posed problem even for varied domains and general data. We present constructive proof of the existence and we show the uniqueness and the continuous dependence with respect to the data of weak solutions of the coupled elastic-gravitational field equations.     

Geomagnetic secular variation and motion of the earth's core

Summary A general theory of a time-dependent magnetic dipole in the earth is discussed. On the basis of the weastward drift of the «equatorial» dipole in the two eccentric dipoles model due toH. G. Macht, the impossibility of the origin of geomagnetic secular variation being in a deep interior of the earth's core is established from the standpoints of the shielding effect and the motions in the core. But the westward drift of the core's top layer relative to the mantle seems to be quite reasonable, even if we take into account the shielding effect of the mantle.     

Propagation of anSH-torque pulse in a three layered solid half-space

Summary The potential functions caused due to anSH-torque pulse from a point source in any of the layers of a three-layered solid half-space have been evaluated in terms of definite integrals, usingCagniard's method. The solution is given as a double infinite series, of which only a finite number of terms contribute to the disturbance at a definite time, and the number of such terms increases with time.     

General circulation of the solar atmosphere from observational evidence

Summary One of the main results of the rotating cylinder experiments ofFultz andHide is that the general flow regime in them is essentially determined by the ratio of the angular velocity of the fluid motions (relative to the cylinder) to that of the cylinder itself. Extending these results to the atmosphere of the sun, leads to the hypothesis that the layer in which spots are imbedded should exhibit a non-axially symmetric pattern, of theRossby type.The fluid motions, characteristic of such a general circulation pattern, are mainly along spherical surfaces, and have a wavelike (eddy) appearance similar to the planetary waves in the upper troposphere of the terrestrial atmosphere. These eddies transport momentum along these spherical surfaces from regions of relatively lower angular velocity to regions of higher velocity. Tracers (e.g., sunspots) imbedded is such a flow would show a correlation between their proper motions in latitude and longitude, such that spots moving equatorward will tend to have larger longitudinal motions (toward the west limb), and vice versa.Analysis of ten years (1935 to 1944) of Greenwich spot data shows a consistent, and (statistically) very significant correlation of spot group proper motions, in the proper sense. These results provide strong support for the existence of large-scale waves which are some modest fraction of the solar circumference, but larger than the sunspot groups. Moreover, these waves transport angular momentum (up the gradient of angular velocity) toward the equatorial regions from higher latitudes across at least the entire sunspot zone. It is not known, however, whether these eddies are the primary (or only) source of momentum to maintain the equatorial acceleration of the sun. However, if this source were shut off, and all other processes continued unabated, this layer of the sun between latitudes ±20° would reach solid rotation in about 51/2 rotations.Because this eddy transport of momentum is counter to the gradient of angular velocity, there is an implied transformation of the kinetic energy of the eddies into the kinetic energy of the mean east-west flow. Of possibly even more interest, however, might be the possibility of transfers of kinetic energy between eddies of all different scale sizes extending down the entire spectrum to include sunspot groups and the spots themselves. Moreover, some eddy size(s) in this layer is likely to be primarily responsible for a conversion of potential to kinetic energy.A result of subsidiary interest is the systematically higher value of solar rotation (at all latitudes) derivable from this data, which includes all spots which survive for at least two days. In contrast to the work of previous authors who used only long-lived spots, the result obtained when many small spots are used, indicates perhaps a variation of the rotation rate with height in the solar atmosphere.The results provide no evidence to indicate the existence of significant meridional circulations (latitudinal driffs).     

Probability of actuating a seismic shutoff device

Automatic seismic shutoff devices are used to reduce the risk to gas and liquid distribution systems from earthquakes. In the USA, the gas shutoff devices are tested and certified according to the American Society of Civil Engineers' Standard ASCE 25. During tests, devices are shaken by simple harmonic (sinusoidal) motions of different frequencies and checked for actuation. Because earthquake motions are not sinusoidal, the amplitude of earthquake motions that will actuate these devices is not clearly understood. This paper determines the probability of actuating devices by earthquake motions of different amplitudes. The probability of actuation increases with increase in the resultant peak horizontal ground acceleration (PGA). The probability of actuation is 50% for PGA = 0.23g and 90% for PGA = 0.31g, where g = 9.81 m/s² = acceleration due to gravity. On a ‘stiff soil’ site in San Francisco, CA, the mean recurrence interval of actuation is 51 years. On a similar site in Boston, MA, the mean recurrence interval of actuation is 3000 years. ASCE 25 compliant devices are actuated by high frequencies in ground motions. There is greater uncertainty in the actuation of these devices by ground motions that are damaging to very flexible systems. Copyright © 2016 John Wiley & Sons, Ltd.     

Effect of seawater on incident plane P and SV waves at ocean bottom and engineering characteristics of offshore ground motion records off the coast of southern California,USA

The effect of seawater on vertical ground motions is studied via a theoretical method and then actual offshore ground motion records are analyzed using a statistical method. A theoretical analysis of the effect of seawater on incident plane P and SV waves at ocean bottom indicate that on one hand, the affected frequency range of vertical ground motions is prominent due to P wave resonance in the water layer if the impedance ratio between the seawater and the underlying medium is large, but it is greatly suppressed if the impedance ratio is small; on the other hand, for the ocean bottom interface model selected herein, vertical ground motions consisting of mostly P waves are more easily affected by seawater than those dominated by SV waves. The statistical analysis of engineering parameters of offshore ground motion records indicate that:(1) Under the infl uence of softer surface soil at the seafl oor, both horizontal and vertical spectral accelerations of offshore motions are exaggerated at long period components, which leads to the peak spectral values moving to a longer period.(2) The spectral ratios(V/H) of offshore ground motions are much smaller than onshore ground motions near the P wave resonant frequencies in the water layer; and as the period becomes larger, the effect of seawater becomes smaller, which leads to a similar V/H at intermediate periods(near 2 s). These results are consistent with the conclusions of Boore and Smith(1999), but the V/H of offshore motion may be smaller than the onshore ground motions at longer periods(more than 5 s).     

On the energy flux of Hide waves

Summary There are two ways to define the energy flux for Hide waves, one from its equations of motion and the other by multiplying its mean energy density by group velocity and these lead to different results FollowingLonguet-Higgins we point out that this is due to certain arbitrariness in the definition of the energy flux and that their difference is just a solenoidal vector.Dedicated to the memory of Dr.V. A. Wynne, 1945–1972.     

Influence of the vertical motion field on ozone concentration in the stratosphere

Computations of the mean meridional motion field in the stratosphere are applied to ozone distributions to evaluate the associated ozone concentration changes. These changes are compared with those produced by photochemical and quasi-horizontal eddy processes. For the period January–April 1964 there is a cooperative action between the mean and eddy motions with mean subsidence in middle latitudes supplying ozone to be carried polawards and equatorwards by quasi-horizontal eddy processes. At low latitudes mean horizontal motions offset the eddy transport while at high latitudes mean rising motion is the offsetting term. The mean ozone flux through 50 mb, 3.5×10²⁹ molecules sec^–1, is comparable with the fluxes evaluated by other techniques.The spring maximum is thought to be due to a modulation of the energy supply to the stratospheric eddies which, in turn, force the mean motions. Longer-term changes are to be expected; for example during Ice Ages when increased tropospheric eddy activity is anticipated there should be higher total ozone.     

On the impossibility of mean-field dynamos with some spherical symmetry of the motions

Abstract

A spherical mean-field dynamo model is considered in which both the mean motion and the mean electro-motive force due to fluctuating motions show some spherical symmetry. It is shown that under some reasonable assumptions the magnetic field is bound to decay to zero.     

High-frequency bottom-pressure and acoustic variations in a sea strait: internal wave turbulence

During a period of 3?days, an accurate bottom-pressure sensor and a four-beam acoustic Doppler current profiler (ADCP) were mounted in a bottom frame at 23?m in a narrow sea strait with dominant near-rectilinear tidal currents exceeding 1?m?s^?1 in magnitude. The pressure record distinguishes small and short surface waves, wind- and ferry-induced near-surface turbulence and waves, large turbulent overturns and high-frequency internal waves. Typical low-frequency turbulent motions have amplitudes of 50?N?m^?2 and periods of about 50?s. Such amplitudes are also found in independent estimates of non-hydrostatic pressure using ADCP data, but phase relationships between these data sets are ambiguous probably due to the averaging over the spread of the slanted acoustic beams. ADCP's echo amplitudes that are observed in individual beams show much better phase correspondence with near-bottom pressure, whether they are generated near the surface (mainly air bubbles) or near the bottom (mainly suspended sediment). These 50-s motions are a mix of turbulence and internal waves, but they are not due to surface wave interactions, and they are not directly related to the main tidal flow. Internal waves are supported by stratification varying between extremely strong thin layer and very weak near-homogeneous stratification. They are driven by the main flow over 2-m amplitude sand-wave topography, with typical wavelengths of 150?m.     

On non-artesian groundwater flow

Summary The present paper deals with certain problems of non-artesian aquifers, with special reference to non-steady flow due to replenishment from the ground surface. Solutions are obtained by direct integration ofBoussinesq's differential equation (transformed), taking also the inclination of the groundwater surface and the slope of the bottom of the aquifer into account. — The study has some bearing on hydrological investigations. For example, it may be desired to find the location of a moving groundwater divide between two water courses, or to determine the quantity of groundwater passing between them. Problems of this nature are also often met with in water supply (artificial infiltration, etc.) and other fields of engineering practice.     

Convection-driven zonal flows in the major planets

The major planets produce heat flux from their interiors that is comparable to the radiative flux they receive from the sun. The dynamics of convection flows carrying the heat flux are discussed, and the dominating effect of the Coriolis force is demonstrated. The characteristic high-velocity jets in the atmospheres of Jupiter and Saturn can be explained on the basis of Reynolds stresses generated by the fluctuating convective motions. A simple annulus model, which elucidates the more complex mathematical analysis of the spherical case given in an earlier paper (Busse, 1983), is considered in detail. Various aspects of the observational evidence are discussed in relation to the model.     

Depression in the low-latitude horizontal intensity due to quiet-time ring current

Summary The external field due to plasma within the magnetosphere has been computed as a function ofA _p, which is a measure of solar wind velocity, for very quiet to slightly disturbed conditions using mean daily horizontal intensity from 1932 to 1968 at Alibag. The intensity, corrected for secular change and reduced to a common epoch, showed initially a small increase withA _p followed by a steady depression with further increase in the index. ForA _p7.5, which is representative of conditions over the 33-hour interval during which data relating to low-energy protons were acquired and used byHoffman andBracken [4]²) to compute current distributions, the decrease, computed here from surface data, is 6 . This is in goodagreement with the southward directed field of the quiet-time proton belt 9±5 obtained byHoffman andBracken.     

Pressure wind adjustment relationships during a multilevel primitive equation prediction process using tropical atmospheric data

Summary In numerical weather forecasting process, with primitive equations, the wind and pressure fields mutually adjust to each other until some form of balance is achieved. The type of balance so achieved by the mass and wind fields during the numerical integration of the primitive equations governing atmospheric motions is not knowna priori. This is particularly so in the case of tropical regions where the pressure wind adjustment laws prevailing in a tropical atmosphere are not well understood.In this study we perform a systematic investigation of the pressure wind adjustment relations during a numerical integration of the primitive equations governing atmospheric motions in a tropical atmosphere. Therefore, a two-day prediction experiment is carried out using the Florida State University Tropical Prediction (FSU) Model (Krishnamurti, 1969;Krishnamurti,et al. 1973;Kanamitsu, 1975). The 200 mb predicted motion (u, v) and height (z) fields are then extracted at 0, 12, 24, 36 and 48 hours of forecast time. Using these motion (u, v) fields three other 200 mb height (z) fields were computed from the inverse nonlinear, linear and quasigeostrophic balance equations. Each of these three diagnostic heights for the 200 mb pressure surface were compared with the respective 200 mb heights obtained from the Florida State University Tropical Preciction Model. The comparison is done by computing the root-mean-square differences between the predicted 200 mb height fields and each of the three 200 mb heights obtained from the inverse non-linear, linear and quasigeostrophic balance equations. The results show that the root-meansquare differences between thez fields from the FSU model and those obtained from the non-linear and linear balance equations lie within the ranges 23 to 44 and 25 to 50 metres respectively. The root-mean-square differences between the predicted heights and the heights computed from the quasigeostrophic balance equation lie in the range 54 to 62 metres. These root-mean-square differences are of significant magnitude since large-scale disturbances in the tropical atmosphere are associated with rather small pressure changes.The variations of these root-mean-square differences as one moves from one forecast time to another exhibit no clear increasing or decreasing trend. In fact the variations appear somewhat random. This rather unsystematic time variation of the root-mean-square differences is a manifestation of the constant changes of the physics in the model as different weather systems evolve in the course of the forecasting process. It seems therefore that the pressure-wind adjustments that take place during a numerical integration of the model equations are of complex nature and cannot simply be approximated by simple diagnostic relations like the ones used in this study.Most of this work was done while the author was at the Florida State University, tallahassee, USA.     

Magnetic buoyancy instabilities incorporating rotation

Abstract

Recent calculations suggest that the bulk of the solar toroidal field may be stored in a thin, convectively stable region situated between the convection zone proper and the radiative zone. Determining the stability properties of such a field is therefore important with implications for both the generation and escape of magnetic flux. The plane layer, linear stability analysis of Hughes (1985) is extended to incorporate the effects of uniform rotation. Detailed studies are made of interchange, or “axisymmetric” modes and of undular, or wavelike, motions, considering modes of both low and high frequency. The force due to rotation acts to constrain the fluid motions, a feature which is strongly stabilizing for direct modes, but can, in certain circumstances, be destabilizing for oscillatory motions.

For the interchange modes we show that the instability discussed at length by Hughes (1985), driven by fields increasing with height, is still present and indeed may be enhanced by rotational effects. We also study the more conventional instabilities, discussing the transformation between direct and oscillatory modes and considering in detail some peculiar properties of the oscillatory instabilities.

The more relevant instabilities in an astrophysical context are likely to be undular modes. Previous studies of low frequency modes driven by top heavy field gradients are extended to consider modes of various frequencies for a wide range of parameter values. Of particular interest is the occurrence of two distinct modes of instability for bottom heavy field gradients. We also exhibit some of the peculiar stability boundaries which can result when none of the competing influences in the problem is dominant.