Accuracy of the kinematic wave and diffusion wave approximations for time-independent flows with infiltration included

Errors in the kinematic wave and diffusion wave approximation for time-independent (or steady-state) cases of channel flow with infiltration were derived for three types of boundary conditions: zero flow at the upstream end, and critical flow depth and zero depth gradient at the downstream end. The diffusion wave approximation was found to be in excellent agreement with the dynamic wave approximation, with errors of less than 1·4% for KF²₀≥7·5, and up to 14% for KF²₀≤0·75 for the upstream boundary condition of zero discharge and finite depth, where K is the kinematic wave number and F₀ is the Froude number. The kinematic wave approximation was reasonably accurate except at the channel boundaries and for small values of KF²₀ (≤1). The accuracy of these approximations was significantly influenced by the downstream boundary, both in terms of the magnitude of the error and the segment of the channel reach for which these approximations would be applicable. © 1998 John Wiley & Sons, Ltd.     

ACCURACY OF KINEMATIC WAVE AND DIFFUSION WAVE APPROXIMATIONS FOR TIME-INDEPENDENT FLOW WITH MOMENTUM EXCHANGE INCLUDED

Errors in the kinematic wave and diffusion wave approximation for time-independent (or steady-state) cases of channel flow with momentum exchange included were derived for three types of boundary conditions: zero flow at the upstream end, and critical flow depth and zero depth gradient at the downstream end. The diffusion wave approximation was found to be in excellent agreement with the dynamic wave approximation, with errors of less than 1% for KF²₀≥7·5 and up to 12% for KF²₀≤0·75 for the upstream boundary condition of zero discharge and finite depth, where K is the kinematic wave number and F₀ is the Froude number. The kinematic wave approximation was reasonably accurate except at the channel boundaries and for small values of KF²₀ (≤1). The accuracy of these approximations was significantly influenced by the downstream boundary condition both in terms of the error magnitude and the segment of the channel reach for which these approximations would be applicable. © 1997 by John Wiley & Sons, Ltd.     

The neural network approximation method for solving multidimensional nonlinear inverse problems of geophysics

The iterative approximation neural network method for solving conditionally well-posed nonlinear inverse problems of geophysics is presented. The method is based on the neural network approximation of the inverse operator. The inverse problem is solved in the class of grid (block) models of the medium on a regularized parameterization grid. The construction principle of this grid relies on using the calculated values of the continuity modulus of the inverse operator and its modifications determining the degree of ambiguity of the solutions. The method provides approximate solutions of inverse problems with the maximal degree of detail given the specified degree of ambiguity with the total number of the sought parameters ~n × 10³ of the medium. The a priori and a posteriori estimates of the degree of ambiguity of the approximated solutions are calculated. The work of the method is illustrated by the example of the three-dimensional (3D) inversion of the synthesized 2D areal geoelectrical (audio magnetotelluric sounding, AMTS) data corresponding to the schematic model of a kimberlite pipe.     

A new approximation of the velocity-depth distribution and its application to the computation of seismic wave fields

Summary A new approximation of the velocity-depth distribution in a vertically inhomogeneous medium is suggested. This approximation guarantees the continuity of velocity and of its first and second derivatives and does not generate false low-velocity zones. It is very suitable for the computations of seismic wave fields in vertically inhomogeneous media by ray methods and its modifications, as it removes many false anomalies from the travel-time and amplitude-distance curves of seismic body waves. The ray integrals can be evaluated in a closed form; the resulting formulae for rays, travel times and geometrical spreading are very simple. They do not contain any transcendental functions (such asln (x) orsin ^–1, (x)) like other approximations; only the evaluation of one square root and of certain simple arithmetic expressions for each layer is required. From a computational point of view, the evaluation of ray integrals and of geometrical spreading is only slightly slower than for a system of homogeneous parallel layers and even faster than for a piece-wise linear approximation.     

Energy spectra associated with long's model for steady finite-amplitude flow over orography

Abstract

The process of wave steepening in Long's model of steady, two-dimensional stably stratified flow over orography is examined. Under conditions of the long-wave approximation, and constant values of the background static stability and basic flow, Long's equation is cast into the form of a nonlinear advection equation. Spectral properties of this latter equation, which could be useful for the interpretation of data analyses under mountain wave conditions, are presented. The principal features, that apply at the onset of convective instability (density constant with height), are:

i) a power spectrum for available potential energy that exhibits a minus eight-thirds decay, in terms of the vertical wavenumber k _z ^-;

ii) a rate of energy transfer across the spectrum that is inversely proportional to the wavenumber for large k _z ^-;

iii) an equipartition between the kinetic energy of the horizontal motion and the available potential energy, under the longwave approximation, although all the disturbance energy is kinetic at the point where convective instability is initiated. It is also shown that features i) and ii) apply to more general conditions that are appropriate to Long's model, not just the long-wave approximation. Application to fully turbulent flow or to conditions at the onset of shearing instability are not considered to be warranted, since the development only applies to conditions at the onset of convective instability.     

Detecting Stochastic Behaviour and Scaling Laws in Time Series of Geomagnetic Daily Means

—We use advanced methods to extract quantitative time dynamics from geomagnetic signals. In particular we analyse daily geomagnetic time series measured at three stations in Norway. The dynamics of geomagnetic measurements has been investigated using autoregressive models. The procedure is based on two forecasting approaches: the global autoregressive approximation and the local autoregressive approximation. The first technique views the data as a realisation of a linear stochastic process, whereas the second considers them as a realisation of a deterministic process, supposedly non-linear. The comparison of the predictive skill of the two techniques is a strong test to discriminate between low-dimensional chaos and stochastic dynamics. Our findings suggest that the physical system governing the phenomena is characterised by a stochastic dynamics, and the process could be described by numerous degrees of freedom. We also investigated the kind of stochasticity of the geomagnetic signals, analysing the power spectrum density. We identify a power law P(?)∝?^-α, with the scaling exponent α which is a typical fingerprint of irregular processes. In this analysis we use the Higuchi method, which presents an interesting relationship between the fractal dimension D and the spectral power law scaling index α.     

The viscous damping of internal waves on the rotating earth

Summary A formula is derived for the spatial attenuation of the two possible modes of oscilation of a two-layer, rotating system. If the long-wave approximation is made, it is found with reference to the internal mode that this formula disagrees with an earlier result obtained byRattray [8]²).     

Annual heat balance and equilibrium temperature of Lake Aegeri,Switzerland

The mean heat budget of Lake Aegeri, Switzerland, is 950 MJ·m^–2, comparable to that of neighbouring lakes. The annual variation in the net heat flux can be adequately described using a six-term heat balance equation based on 12 years of monthly mean meteorological and surface temperature data. Although the magnitude of the net heat flux is dominated by the radiative terms of the equation, the one-month backward shift of the net flux and total heat content extrema from the solstices and equinoxes respectively is due to the phase shift of the non-radiative with respect to the radiative terms. A linear approximation was used to express the net heat flux in terms of a heat exchange coefficient and an equilibrium temperature. The former varies from 17 to 28 W·m^–2·K^–1 in the course of a year; fluctuations in the latter are found to depend mainly on fluctuations in cloud cover and relative humidity, whilst the effect of fluctuations in air temperature and wind speed is slight.     

Difficulties in the application of magnetic field gradient analysis to induction studies

Geomagnetic variation data recorded across an array of magnetometer stations in northern England have been used to compute an inductive response function. The function, if successfully determined, is appropriate for sounding vertical conductivity structures on a regional scale. The method employed is the horizontal spatial gradient method using station averaging and a linear approximation. It is found that, for the data under consideration, local and lateral anomalous fields exist in all three components at periods ranging from 10 to 10⁴ s. The presence of such anomalous fields invalidates the procedures used to estimate the regional field, whose characteristics are required to determine successfully the inductive response function.     

Snow in Lebanon: a preliminary study of snow cover over Mount Lebanon and a simple snowmelt model / Etude préliminaire du couvert neigeux et modèle de fonte des neige pour le Mont Liban

Abstract

The physical properties of snow, including apparent density, snow cover distribution and snowmelt in the Nahr El Kelb basin (Mount Lebanon), were studied in order to design a simple empirical snowmelt model. In February 2001, snow covered an area of 1600 km² on Mount Lebanon, representing a water equivalent of 1.1 x 10⁹ m³. The snow surface area was calculated by combining TM5 images with a digital elevation model, and field observations made every three days, from 1400 to 2300 m altitude. The depletion of snow cover was measured from the end of December 2000 to the end of June 2001. The snowmelt was measured from surface depletion on a degree-day basis. A simple model relating the daily snowmelt to the product of wind speed and average positive daily air temperature, is presented and discussed. For Mount Lebanon, this model gave a better approximation of snowmelt than a simple degree-day model.     

A rational explanation of cross-profile morphology for glacial valleys and of glacial valley development

The fact that the cross-profile of the glacial valley could be well approximated by parabolas (Y = aX^b, b = 2.0) is explained by the variation principle, assuming that the glacier erosion works towards minimizing thefriction between ice and bedrock. The variation principle proves that the ideal or fully-developed morphology of the glacial valley should be a catenary, the curve which a chain hanging from two fixed points forms. Maclaurin's series expansion of the catenary equation shows that a parabola is a very good approximation of the catenary; hence, the good approximation of the cross-profile by parabolas. Different catenaries are generated by changing the form ratio (depth/rim width) and are then approximated by Y = aX^b by the method of last-squares. The b values obtained become only fractionally larger than 2.0 with invreasing form ratios of up to 1.0, indicating that b values would range, in practice, between 1.0 and about 2.0 Two types of trend in the relationship between b values and the form ratio were obtained from several glaciers. For one type the b value becomes larger with increasing form ratios, and for the other the opposite. The first type is called the Rocky Mountain model after its source of data and represents overdeepening of the glacial valley development. The second type is caalled the Patagonia-Antarctica model, representing a widening, instead of a deepening, process of development. These differences are attributed to the nature of the glaciers which produced these valleys, i.e. alpine glaciers and continental ice sheets.     

Convection and lunar thermal history

The effect of solid convection on the thermal evolution of the Moon is explored for a variety of viscosities, radioactive differentiation efficiencies and initial temperature profiles. Convective heat flux in the models is calculated using an empirical relation derived from the results of laboratory experiments and numerical solutions of the Navier-Stokes equations. The method retains the spherically symmetric approximation and, therefore, greatly facilitates numerical calculations.Results show that even though solid convection may determine the thermal state of the lunar interior, it does not necessarily produce a quasi-steady thermal balance between heat sources and surface loss. An imbalance persists, due to the cooling and growth of the nonconvecting lithosphere. The state of the lithosphere is sensitive to the efficiency of heat source redistribution, while that of the convecting interior depends primarily on rheology. Convecting models have viscosities of 10²¹–10²² cm²s^?1 in their interiors; the central temperature must be above 1100°C. Convection occurring within the first billion years after formation could have led to mare flooding by magma produced in hot zones of convection cells. However, it cannot be shown from model calculations alone that solid convection must have dominated lunar thermal history.     

The effect of the electron sound speed on wave propagation in the ionospheric plasma

In this study, the effect of the electron sound speed on the extraordinary wave propagation is calculated without an approximation for either collisional or collisionless cases in the ionospheric plasma by using the real geometry of the Earth’s magnetic field for the Northern Hemisphere. It is observed that there is no remarkable effect on the propagation of the extraordinary wave, especially at reflection altitudes. But it is also observed that the magnitudes of k ² (the square of the wave number) have changed every season, and the phase velocity of wave in warm ionospheric plasma has increased.     

Using geophysical surveys to test tracer‐based storage estimates in headwater catchments

Hydrogeophysical surveys were carried out in a 3.2 km² Scottish catchment where previous isotope studies inferred significant groundwater storage that makes important contributions to streamflow. We used electrical resistivity tomography (ERT) to characterize the architecture of glacial drifts and make an approximation of catchment‐scale storage. Four ERT lines (360–535 m in length) revealed extensive 5–10 m deep drift cover on steeper slopes, which extends up to 20–40 m in valley bottom areas. Assuming low clay fractions, we interpret variable resistivity as correlating with variations in porosity and water content. Using Archie's Law as a first approximation, we compute likely bounds for storage along the ERT transects. Areas of highest groundwater storage occur in valley bottom peat soils (up to 4 m deep) and underlying drift where up to 10 000 mm of precipitation equivalent may be stored. This is consistent with groundwater levels which indicate saturation to within 0.2 m of the surface. However, significant slow groundwater flow paths occur in the shallower drifts on steeper hillslopes, where point storage varies between ~1000 mm–5000 mm. These fluxes maintain saturated conditions in the valley bottom and are recharged from drift‐free areas on the catchment interfluves. The surveys indicate that catchment scale storage is >2000 mm which is consistent with tracer‐based estimates. Copyright © 2016 John Wiley & Sons, Ltd.     

On the accuracy of the bathymetry-generated gravitational field quantities for a depth-dependent seawater density distribution

In geophysical studies investigating the lithosphere structure, the gravitational field generated by the ocean density contrast (i.e., bathymetry-generated gravitational field) represents a significant amount of the signal to be modelled and subsequently removed from the Earth’s gravity field. The ocean density contrast is typically calculated as the difference between the mean density values of the Earth’s crust and seawater. The approximation of the actual seawater density distribution by its mean value yields relative errors up to about 2% in computed quantities of the gravitational field. To reduce these errors, a more realistic model of the seawater density distribution is utilized based on the analysis of existing oceanographic data of salinity, temperature, and pressure (depth). We study the accuracy of the bathymetry-generated gravitational field quantities formulated for a depth-dependent model of the seawater density distribution. This density distribution approximates the seawater density variations due to an increasing pressure with depth, whereas smaller lateral density variations caused by salinity, temperature, and other oceanographic factors are not taken into consideration. The error analysis reveals that the approximation of the seawater density by the depth-dependent density model reduces the maximum errors to less than 0.6%. The corresponding depth-averaged errors are below 0.1%. The depth-dependent seawater density model is further facilitated in expressions for computing the bathymetry-generated gravitational field quantities by means of the spherical bathymetric (ocean bottom depth) functions. The numerical realization reveals large differences in the results obtained with and without consideration of the depth-dependent seawater density distribution. The maxima of absolute differences reach 201 m²/s² and 16.5 mGal in computed values of the potential and attraction, respectively. The application of the depth-dependent seawater density model thus significantly improves the accuracy in the forward modelling of the bathymetric gravitational field quantities.     

A note on edge waves on a flat shelf

Abstract

A variational approximation to the dispersion relation for trapped waves on a flat shelf of depth h ₁, bounded internally by a vertical coast and externally by a semi-infinite ocean of depth h ₂>h ₁, is obtained through an integral-equation formulation that accounts for all of the non-propagated modes that are excited at the discontinuity in depth (the conventional formulation of the edge-wave problem allows only for the propagated mode on the shelf and the dominant, non-propagated mode in the deep water). Coriolis effects are neglected. The exact result in the limit ω² h ₂/g↓0 (ω = angular frequency) is obtained by conformal mapping and compared with the variational approximation, which proves to be quite accurate over the entire range 1>h ₂/h ₁>x. The effects of the higher-order, non-propagated modes are found to be small for the long waves observed over the Southern California shelf by Snodgrass, Munk and Miller (1962).     

Вычисление первой и второй вертикальных производных силы тяжести

Summary Following Molodensky's suggestions anomalies of the vertical gradient of gravity were used to achieve a greater accuracy in the determination of the figure of the Earth by gravimetrical methods. The existing methods of computing this quantity do not take into account inclinations of the physical surface of the Earth. Using the Laplace equation, the second derivative ∂² T/∂v ² (1) of the disturbing potentialT is expressed by the second derivatives ofT along the tangentsτ ₁ andτ ₂ to the physical surface of the Earth in mutually perpendicular planes and by the derivatives of gravity anomalies (2). The derivatives ∂² T/∂τ ₁ ² and ∂² T/∂τ ₂ ² have been determined using the Molodensky method [4] of solving his integral equation for the single layer density. In the zero approximation, the Noumerov formula [2] was obtained; however, the results obtained using this formula should be referred to the physical surface of the Earth, not to the Listing geoid. The correction of the first approximation is given by formula (16). The second vertical derivative of gravity anomalies can be determined using the expression (20).      

Approximating the Distribution of Pareto Sums

Heavy tailed random variables (rvs) have proven to be an essential element in modeling a wide variety of natural and human-induced processes, and the sums of heavy tailed rvs represent a particularly important construction in such models. Oriented toward both geophysical and statistical audiences, this paper discusses the appearance of the Pareto law in seismology and addresses the problem of the statistical approximation for the sums of independent rvs with common Pareto distribution F(x)=1 – x^– for 1/2 < < 2. Such variables have infinite second moment which prevents one from using the Central Limit Theorem to solve the problem. This paper presents five approximation techniques for the Pareto sums and discusses their respective accuracy. The main focus is on the median and the upper and lower quantiles of the sums distribution. Two of the proposed approximations are based on the Generalized Central Limit Theorem, which establishes the general limit for the sums of independent identically distributed rvs in terms of stable distributions; these approximations work well for large numbers of summands. Another approximation, which replaces the sum with its maximal summand, has less than 10% relative error for the upper quantiles when < 1. A more elaborate approach considers the two largest observations separately from the rest of the observations, and yields a relative error under 1% for the upper quantiles and less than 5% for the median. The last approximation is specially tailored for the lower quantiles, and involves reducing the non-Gaussian problem to its Gaussian equivalent; it too yields errors less than 1%. Approximation of the observed cumulative seismic moment in California illustrates developed methods.     

Hard X rays of relativistic electrons accelerated in solar flares

The direction and polarization degree of hard X rays (HXRs) in solar flares are studied. The continuous injection of relativistic electrons, which is implemented in powerful flares, is considered. The stationary relativistic kinetic equation is studied by using the method of expansion in terms of the Legendre polynomial and by integrating the equations for the expansion coefficients. The HXR characteristics are calculated using the bremsstrahlung relativistic cross-section for different angular and energetic electron distributions in the acceleration region. A high linear polarization degree of HXRs (??35%) has been obtained for narrow (??cos⁶??) beams of electrons with a soft spectrum (??E ^?6); the polarization degree decreases with increasing quanta energy, whereas the directivity of a high-energy emission increases. This effect is absent for a nonrelativistic approximation. The considered model is applied to one of the most powerful flares in cycle 23, registered on October 28, 2003. The measured polarization degree values at relativistic energies (0.2?C0.4 and 0.4?C1 MeV) agree with the results achieved in the considered model when the electron energy spectrum index (?? = 2.5), angular distribution part (??cos⁶??), and the spectrum cutoff energy (E _max = 1.3 MeV) were specified.