Penetrative convection in rotating fluids: A model for the base of stellar convection zones

Abstract

To model penetrative convection at the base of a stellar convection zone we consider two plane parallel, co-rotating Boussinesq layers coupled at their fluid interface. The system is such that the upper layer is unstable to convection while the lower is stable. Following the method of Kondo and Unno (1982, 1983) we calculate critical Rayleigh numbers R_c for a wide class of parameters. Here, R_c is typically much less than in the case of a single layer, although the scaling R_c～T^2/3 as T → ∞ still holds, where T is the usual Taylor number. With parameters relevant to the Sun the helicity profile is discontinuous at the interface, and dominated by a large peak in a thin boundary layer beneath the convecting region. In reality the distribution is continuous, but the sharp transition associated with a rapid decline in the effective viscosity in the overshoot region is approximated by a discontinuity here. This source of helicity and its relation to an alpha effect in a mean-field dynamo is especially relevant since it is a generally held view that the overshoot region is the location of magnetic field generation in the Sun.     

Estimation of rainfall elasticity of streamflow in Australia

Abstract

Estimates of rainfall elasticity of streamflow in 219 catchments across Australia are presented. The rainfall elasticity of streamflow is defined here as the proportional change in mean annual streamflow divided by the proportional change in mean annual rainfall. The elasticity is therefore a simple estimate of the sensitivity of long-term streamflow to changes in long-term rainfall, and is particularly useful as an initial estimate of climate change impact in land and water resources projects. The rainfall elasticity of streamflow is estimated here using a hydrological modelling approach and a nonparametric estimator. The results indicate that the rainfall elasticity of streamflow (? _P ) in Australia is about 2.0–3.5 (observed in about 70% of the catchments), that is, a 1% change in mean annual rainfall results in a 2.0–3.5% change in mean annual streamflow. The rainfall elasticity of streamflow is strongly correlated to runoff coefficient and mean annual rainfall and streamflow, where streamflow is more sensitive to rainfall in drier catchments, and those with low runoff coefficients. There is a clear relation-ship between the ? _P values estimated using the hydrological modelling approach and those estimated using the nonparametric estimator for the 219 catchments, although the values estimated by the hydrological modelling approach are, on average, slightly higher. The modelling approach is useful where a detailed study is required and where there are sufficient data to reliably develop and calibrate a hydrological model. The nonparametric estimator is useful where consistent estimates of the sensitivity of long-term streamflow to climate are required, because it is simple to use and estimates the elasticity directly from the historical data. The nonparametric method, being model independent, can also be easily applied in comparative studies to data sets from many catchments across large regions.     

Coherency of rupture and seismic spectrum

In considering the seismic spectrum, one of the methods to incorporate irregularity of fault motion statistically is to introduce the concept of coherency of fracture. In a classic paper,Aki (1967) investigated the scaling law of seismic spectrum on the basis of a statistical model in which an exponentially decaying function is fitted to the autocorrelation function of the dislocation velocity. It is found, however, thatAki's model does not necessarily express irregular fault motion, but corresponds to a smooth dislocation. We show that an analytical function of dislocation velocity gives the same autocorrelation function and the same seismic spectrum as those ofAki's model. In actual fault motion, there is considerable evidence which indicates that the dislocation is not continuous and smooth over the whole fault plane, but is often segmented in several parts. In order to take into consideration this feature we introduce a generalized autocorrelation function of the dislocation velocity in which many coherent fractures smaller than the size of the fault dimension are included. It is shown that the more small-scale coherent fractures, the larger the seismic wave energy in the high frequency range.Kanamori andAllen (1986) showed that a large ratio of seismic wave energy relative to the seismic moment means a large effective stress drop. On the other hand, it is well known that when a fault plane is segmented in several parts, stress drop becomes large (e.g.,Madariaga, 1979;Rudnicki andKanamori, 1981). These two results are fused in our model, because we find that large seismic wave energy is obtained when the fault motion includes small-scale fractures.Kanamori andAllen (1986) also showed that there is a tendency for earthquakes with long repeat times to have a large effective stress drop. Our model implies that a fracture corresponding to earthquakes with long recurrence intervals is more complex, and the strength is large, as also suggested byCao andAki (1986) using a numerical simulation. It should be noted that to the zeroth order, an approximate scaling relation is observed among earthquakes, which means that a large earthquake consists of a relatively large-scale coherent fracture. This fact seems to suggest that the condition of occurrence of a large earthquake is related to the maturing of a source region in which a large coherent fracture becomes feasible.     

The equatorial dynamics of a shallow,homogeneous ocean

Abstract

It is shown that the linear equatorial dynamics of a shallow ocean is characterized by two boundary layers of width γ^? L and γL (γ is the Ekman number of the flow, assumed small, and L is a horizontal dimension of the basin). In the γ^? layer stress in the bottom Ekman layer is comparable to that in the surface Ekman layer. In the γ layer vertical friction is important throughout the depth of the ocean. Should the Rossby number ? be so large as to invalidate a linear theory (? > γ^5/3), then inertial effects become important at a distance ?^2/5 L from the equator. The role played in the circulation of the basin by the non-linear equatorial current first studied by Charney (1960) is shown to be similar to that of the γ layer of the linear theory. Though lateral friction is unimportant in a linear model of the flow, shear layers at the equator are found to be a necessary feature of non-linear flow.     

Design Earthquakes in the U.K.

Three sites in the UK are taken, representative of low, medium and high hazard levels (by UK standards). For each site, the hazard value at 10⁻⁴ annual probability is computed using a generic seismic source model, and a variety of ground motion parameters: peak ground acceleration (PGA), spectral acceleration at 10 Hz and 1 Hz, and intensity. Disaggregation is used to determine the nature of the earthquakes most likely to generate these hazard values. It is found (as might be expected) that the populations are quite different according to which ground motion parameter is used. When PGA is used, the result is a rather flat magnitude distribution with a tendency to low magnitude events (\le 4.5 ML) which are probably not really hazardous. Hazard-consistent scenario earthquakes computed using intensity are found to be in the range 5.8–5.9 ML, which is more in accord with the type of earthquake that one expects to be a worst-case event in the UK. This revised version was published online in July 2006 with corrections to the Cover Date.     

A comparison of two numerical models for the prediction of vibrations from underground railway traffic

Two prediction models for calculating vibration from underground railways are developed: the pipe-in-pipe model and the coupled periodic finite element–boundary element (FE–BE) model.The pipe-in-pipe model is a semi-analytical three-dimensional model that accounts for the dynamic interaction between the track, the tunnel and the soil. The continuum theory of elasticity in cylindrical coordinates is used to model two concentric pipes: an inner pipe to represent the tunnel wall and an outer pipe to represent the surrounding soil. The tunnel and soil are coupled accounting for equilibrium of stresses and compatibility of displacements at the tunnel–soil interface. This method assumes that the tunnel is invariant in the longitudinal direction and the problem is formulated in the frequency–wavenumber domain using a Fourier transformation. A track, formulated as an Euler–Bernoulli beam, is then coupled to this model. Results are transformed to the space domain using the inverse Fourier transform.The coupled periodic FE–BE model is based on a subdomain formulation, where a boundary element method is used for the soil and a finite element method for the tunnel. The Craig–Bampton substructuring technique is used to efficiently incorporate the track in the tunnel. The periodicity of the tunnel is exploited using the Floquet transformation to formulate the track–tunnel–soil interaction problem in the frequency–wavenumber domain and to compute the wave field radiated into the soil.An invariant concrete tunnel, embedded in a homogeneous full space is analyzed using both approaches. The pipe-in-pipe model offers an exact solution to this problem, which is used to validate the coupled periodic FE–BE model. The free field response due to a harmonic load in the tunnel is predicted and results obtained with both models are compared. The advantages and limitations of both models are highlighted. The coupled periodic FE–BE model has a greater potential as it can account for the complex periodic geometry of the tunnel and the layering in a soil medium. The effect of coupling a floating slab to the tunnel–soil system is also studied with both models by calculating the insertion gain.     

The nonlinear stabilization of a zonal shear flow instability

Abstract

The development of initially small perturbations in a weakly supercritical zonal shear flow on a β-plane is studied. Two different scenarios of evolution are possible. If the supercriticality is sufficiently small, the growth of a perturbation is stopped in the viscous critical layer regime; for this case the evolution equation (corrected by the inclusion of a quintic nonlinearity) is derived. At greater supercriticality the nonlinearity cannot stop the growth of the perturbation in a linear (viscous or unsteady) critical layer regime, and the evolution is more complicated. Transition to a nonlinear critical layer regime leads to a reduction in the growth rate and to a slowing (but not a stopping) of the increase in amplitude, A. These are connected to the formation of a plateau (S=constant) of width L=O(A ^½) in the profile of absolute vorticity, S. Careful analysis reveals that the growth in amplitude ceases only when the whole instability domain (where the slope of unperturbed S-profile is positive) becomes covered again by the plateau.     

Point dilution tests to calculate groundwater velocity: an example in a porous aquifer in northeast Italy

ABSTRACT

The point dilution test is a single-well technique for estimating horizontal flow velocity in the aquifer surrounding a well. The test is conducted by introducing a tracer into a well section and monitoring its decreasing concentration over time. When using a salt tracer, the method is easy and inexpensive. Traditionally, the horizontal Darcy velocity is calculated as a function of the rate of dilution and is based on the simple assumption that the decreasing tracer concentration is proportional both to the apparent velocity into the test section and to the Darcy velocity in the aquifer. In this article, an alternative approach to analyse the results of point dilution tests is proposed and verified using data acquired at a test site in the middle Venetian plain, northeast Italy. In this approach, the one-dimensional equilibrium advection–dispersion equation is inverted using the CXTFIT model to estimate the apparent velocity inside the test section. Analysis of the field data obtained by the two approaches shows good agreement between the methods and suggests that it is possible to use the equilibrium advection–dispersion equation to estimate apparent velocity over a wide range of velocities.

Editor D. Koutsoyiannis; Associate editor K. Heal     

Short communication block lanczos method for dynamic analysis of structures

The simple Lanczos method presented in a recent paper by the writers, with application to single vector loads, is extended to include a more general dynamic loading represented as a linear combination of k vectors (load patterns). The result is a set of orthogonal vectors that is used to transform the equations of motion to a banded form, the half-bandwidth of which becomes k + 1. When k is small relative to the number of equations, this approach provides for a very efficient time-stepping solution.     

TRANSIENT EM RESPONSE OF A LARGE LOOP ON A LAYERED GROUND *

The voltage induced in a horizontal loop on a layered ground has been calculated for the case where the loop is excited by a step current and measurements are made during the off-cycle. The expressions derived for a uniform ground show that for large time t the induced voltage E(t) is approximately given by E(t)?— (Ibαμ/20t) (σμ²/t)^3/2 where σ is the conductivity of the ground, μ the permeability, b the loop radius, and I the amplitude of the current step. For small times the corresponding result is E(t)?—Ibμ/2t. When the ground is composed of a number of layers a numerical procedure for calculating the induced voltage is described. The calculated responses of various multilayered structures show that at short times the induced voltage is asymptotic to that produced in the case of a uniform ground of conductivity equal to the top layer. Interference effects in the top layer can lead to anomalous decay curves which may result in the underestimation of the conductivity of a buried layer.     

An analytical solution to multi-component NAPL dissolution equations

This note presents a novel method for determining the changing composition of a multi-component NAPL body dissolving into moving groundwater, and the consequent changes in the aqueous phase solute concentrations in the surrounding pore water. A canonical system of coupled non-linear governing equations is derived which is suitable for representation of both pooled and residual configurations, and this is solved. Whereas previous authors have handled such problems numerically, it is shown that these governing equations succumb to analytical solution. By a suitable substitution, the equations become decoupled, and the problem collapses to a single first-order equation. The final result is expressed implicitly, with time as a function of the number of moles of the least soluble component, m₁. The number of moles of each other component is expressed explicitly in terms of m₁. It is shown that the time-m₁ relationship has a well behaved inverse. An example is given in which the analytic solution is verified against traditional finite difference analysis, and its computational efficiency is shown.     

Convection of geothermal fluids in the timanfaya volcanic area (Lanzarote,Canary Islands)

Summary A mathematical model has been derived to study the superficial thermal anomalies to be found in Lanzarote (605°C at 13 m depth) in association with the convection of geothermal fluids. The model is valid for a wide range of conditions, in particular for those found beneath the Timanfaya volcano (active between 1730 and 1736). Geological and geophysical data suggest that the heat source is related to a cylindrical magma body with a radius of 200±100 m and a top temperature of 850±100°C at a depth of 4±1 km.Energy is transported through fractures by magmatic volatiles and/or by water vapour coming from a deeply located water table: in such a convection system, a fluid flow of 10 l/m² day, which corresponds to a thermal flux of 130 W/m², is sufficient to explain the temperature anomalies observed at the surface. The relationships between gas flow and the surface temperatures, as well as the thermal gradients in the conducting fracture are also discussed.     

Transient Divergent Flow and Transport in an Infinite Anisotropic Porous Formation

When seeking to predict plume geometry resulting from fluid injection through partially penetrating wells, it is common to assume a steady-state spherically diverging flow field. In reality, the flow field is transient. The steady-flow assumption is likely to cause overestimation of injection plume radius since the accommodation of fluid by increases in porosity and fluid density is ignored. In this paper, a transient solution is developed, resulting in a nonlinear ordinary differential equation expressing plume radius as a function of time. It is shown that the problem can be fully described by one type curve. A critical time, t_c, is identified at which the percentage error of the steady-state flow solution compared to the fully dynamic problem is less than 1%. Only for large injection rates and low permeabilities, does t_c become greater than 1 h. Nevertheless, an improved approximate solution is obtained by a simple linearization procedure. The critical time, t_c for the new approximate solution is 0.3% of that required for the steady-state flow solution.     

The deepening of the wind-Mixed layer

Abstract

A simple model is given that describes the response of the upper ocean to an imposed wind stress. The stress drives both mean and turbulent flow near the surface, which is taken to mix thoroughly a layer of depth h, and to erode the stably stratified fluid below. A marginal stability criterion based on a Froude number is used to close the problem, and it is suggested that the mean momentum has a strong role in the mixing process. The initial deepening is predicted to obey

where u. is the friction velocity of the imposed stress, N the ambient buoyancy frequency, and t the time.

After one-half inertial period the deepening is arrested by rotadeon at a depth h = 2^2/4 u.{(Nf)⁺

where f is the Coriolis frequency. The flow is then a “mixed Ekman” layer, with strong inertial oscillations superimposed on it. Three quarters of the mean energy of the deepening layer is found to be kinetic, and only one-quarter potential.

Heating and cooling are included in the model, but stress dominates for time-scales of a day or less. Non-uniform stratification and currents existing prior to the onset of the wind are easily included.

Agreement between the first formula above and laboratory experiments of Kato and Phillips is very satisfactory; the second formula is consistent with observations of Francis and Stommel, though a more thorough test is needed. Oceanic observations in general support the assumption of slab-like mean profiles and direct response of the fluid to local winds.     

A Discrete Wavenumber Boundary Element Method for study of the 3-D response 2-D scatterers

A mathematical formulation of the 2·5D elastodynamic scattering problem is presented and validated. The formulation is a straightforward extension of the Discrete Wave number Boundary Integral Equation Method (DWBIEM) originally proposed by Kawase¹ for 2D scattering problems and subsequently extended to the 3D problem by Kim and Papageorgiou.² It is demonstrated that the Green's function which is appropriate for a boundary formulation of the 2·5D elastodynamic scattering problem is the one corresponding to a unit force moving on a straight line with constant velocity. Such a Green's function is derived in the present study. The formulation may be used to study the wavefields in models of sedimentary deposits (e.g. valleys) or topography (e.g. canyons or ridges) with a 2D variation in structure but obliquely incident plane waves. The advantage of a 2·5D formulation is that it provides the means for calculations of 3D wavefields in scattering problems by requiring a storage comparable to that of the corresponding 2D calculations. © 1998 John Wiley & Sons, Ltd.     

Recent progress in estimating uncertainty in geomagnetic field modeling

Abstract

Recent work pertaining to estimating error and accuracies in geomagnetic field modeling is reviewed from a unified viewpoint and illustrated with examples. The formulation of a finite dimensional approximation to the underlying infinite dimensional problem is developed. Central to the formulation is an inner product and norm in the solution space through which a priori information can be brought to bear on the problem. Such information is crucial to estimation of the effects of higher degree fields at the Core-Mantle boundary (CMB) because the behavior of higher degree fields is masked in our measurements by the presence of the field from the Earth's crust. Contributions to the errors in predicting geophysical quantities based on the approximate model are separated into three categories: (1) the usual error from the measurement noise; (2) the error from unmodeled fields, i.e. from sources in the crust, ionosphere, etc.; and (3) the error from truncating to a finite dimensioned solution and prediction space. The combination of the first two is termed low degree error while the third is referred to as truncation error.

The error analysis problem consists of “characterizing” the difference δz = z—z, where z is some quantity depending on the magnetic field and z is the estimate of z resulting from our model. Two approaches are discussed. The method of Confidence Set Inference (CSI) seeks to find an upper bound for |z—?|. Statistical methods, i.e. Bayesian or Stochastic Estimation, seek to estimate E(δz² ), where E is the expectation value. Estimation of both the truncation error and low degree error is discussed for both approaches. Expressions are found for an upper bound for |δz| and for E(δz² ). Of particular interest is the computation of the radial field, B., at the CMB for which error estimates are made as examples of the methods. Estimated accuracies of the Gauss coefficients are given for the various methods. In general, the lowest error estimates result when the greatest amount of a priori information is available and, indeed, the estimates for truncation error are completely dependent upon the nature of the a priori information assumed. For the most conservative approach, the error in computing point values of B_r at the CMB is unbounded and one must be content with, e.g., averages over some large area. The various assumptions about a priori information are reviewed. Work is needed to extend and develop this information. In particular, information regarding the truncated fields is needed to determine if the pessimistic bounds presently available are realistic or if there is a real physical basis for lower error estimates. Characterization of crustal fields for degree greater than 50 is needed as is more rigorous characterization of the external fields.     

Monitoring and mapping of hydrogen sulphide emissions across an active geothermal field: Rotorua,New Zealand

Hydrogen sulphide (H₂S) is one of a number of gaseous species associated with geothermal activity in the Taupo Volcanic Zone (TVZ), New Zealand. The city of Rotorua is located within Rotorua Caldera in the TVZ and is one of the few urban areas in the world where a large population (>60,000 people) is frequently exposed to geothermal emissions. In order to evaluate the health hazard from long-term exposure to H₂S being emitted from the Rotorua geothermal field, a passive sampler has been developed to measure concentrations of H₂S at many locations across the city simultaneously. In contrast to other passive or pump-based samplers, the sampler is inexpensive, easily mass-manufactured, and involves the reaction of H₂S with silver halide contained in treated photographic paper. H₂S-exposed paper shows a distinct colour change from white to dark brown as H₂S concentrations increase and is sensitive to concentrations between ≪30 and around 1000 ppb. Rotorua city can be divided into three regions—an area of low H₂S concentration in the west, a ‘corridor’ of high concentrations running north–south through the city centre where H₂S is being emitted, and an area of medium concentration to the east which is influenced by the prevailing wind direction, creating a plume from the central corridor. The data give new insight into the subsurface routes of degassing in the Rotorua geothermal field, by showing the surface expression of the main upflow zone and the direction of the conjectured faulting below.     

NON-NORMAL INCIDENCE INVERSION: EXISTENCE OF SOLUTION*

An inverse problem is one in which the parameters of a model are determined from measured seismic data. Important to the solution of inverse problems is the issue of whether or not a solution exists. In this paper we show, in a constructive manner, that a solution does exist to the specific inverse problem of determining the parameters of a horizontally stratified, lossless, isotropic and homogeneous layered system that is excited by a non-normal incidence (NNI) plane wave. Mode conversion between P- and S-waves is included. We develop a seven-step layer-recursive procedure for determining all of the parameters for layer j. These parameters are P-wave and S-wave velocities and angles of incidence, density, thickness, traveltimes, and reflection- and transmission-coefficient matrices. Downward continuation of data from the top of one layer to the top of the next lower layer is an important step in our procedure, just as it is in normal incidence (NI) inversion. We show that, in order to compute all parameters of layer j, we need to (and can) compute some parameters for layer j+ 1. This is a non-causal phenomenon that seems to be necessary in NNI inversion but is not present in NI inversion.     

Gravity profiles across the Cheyenne Belt, a precambrian crustal suture in southeastern Wyoming

Geologic discontinuities across the Cheyenne Belt of southeastern Wyoming have led to interpretations that this boundary is a major crustal suture separating the Archaean Wyoming Province to the north from accreted Proterozoic island arc terrains to the south. Gravity profiles across the Cheyenne Belt in three Precambrian-cored Laramide uplifts show a north to south decrease in gravity values of 50–100 mgal. These data indicate that the Proterozoic crust is more felsic (less dense) and/or thicker than Archaean crust. Seismic refraction data show thicker crust (48–54 km) in Colorado than in Wyoming (37–41 km). We model the gravity profiles in two ways: 1) thicker crust to the south and a south-dipping ramp in the Moho beneath and just south of the Cheyenne Belt; 2) thicker crust to the south combined with a mid-crustal density decrease of about 0.05 g/cm³. Differences in crustal thickness may have originated 1700 Ma ago because: 1) the gravity gradient is spatially related to the Cheyenne Belt which has been immobile since about 1650 Ma ago; 2) the N-S gradient is perpendicular to the trend of gravity gradients associated with local Laramide uplifs and sub-perpendicular to regional long-wavelength Laramide gradients and is therefore probably not a Laramide feature. Thus, gravity data support the interpretation that the Cheyenne Belt is a Proterozoic suture zone separating terrains of different crustal structure. The gravity “signature” of the Cheyenne Belt is different from “S”-shaped gravity anomalies associated with Proterozoic sutures of the Canadian Shield which suggests fundamental differences between continent-continent and island arc-continent collisional processes.     

THE CAGNIARD METHOD IN COMPLEX TIME REVISITED1

The Cagniard-de Hoop method is ideally suited to the analysis of wave propagation problems in stratified media. The method applies to the integral transform representation of the solution in the transform variables (s, p) dual of the time and transverse distance. The objective of the method is to make the p-integral take the form of a forward Laplace transform, so that the cascade of the two integrals can be identified as a forward and inverse transform, thereby making the actual integration unnecessary. That is, the exponent (–sw(p)) is set equal to –sτ, with τ varying from some (real) finite time to infinity. As usually presented, the p-integral is deformed onto a contour on which the exponent is real and decreases to –∞ as p tends to infinity. We have found that it is often easier to introduce a complex variable τ for the exponent and carry out the deformation of contour in the complex τ-domain. In the τ-domain the deformation amounts to ‘closing down’ the contour of integration around the real axis while taking due account of singularities off this axis. Typically, the method is applied to an integral that represents one body wave plus other types of waves. In this approach, the saddle point of w(p) that produces the body wave plays a crucial role because it is always a branch point of the integrand in the τ-domain integral. Furthermore, the paths of steepest ascent from the saddle point are always the tails of the Cagniard path along which w(p) →∞. That is, the image of the pair of steepest ascent paths in the p-domain is a double covering of a segment of the Re τ-axis in the τ-domain. The deformed contour in the p-domain will be the only pair of steepest ascent paths unless the original integrand had other singularities in the p-domain between the imaginary axis and this pair of contours. This motivates the definition of a primary p-domain, i.e. the domain between the imaginary axis and the steepest descent paths, and its image in the τ-domain, the primary τ-domain. In terms of these regions, singularities in the primary p-domain have images in the primary τ-domain and the deformation of contour on to the real axis in the τ-domain must include contributions from these singularities. This approach to the Cagniard-de Hoop method represents a return from de Hoop's modification to Cagniard's original method, but with simplifications that make the original method more tractable and straightforward. This approach is also reminiscent of van der Waerden's approach to the method of steepest descents, which starts exactly the same way. Indeed, after the deformation of contour in the τ-domain, the user has the choice of applying asymptotic analysis to the resulting ‘loop’ integrals (Watson's lemma) or continuing to obtain the exact, although usually implicit, time-domain solution by completing the Cagniard-de Hoop analysis. In developing the method we examine the transformation from a frequency-domain representation of the solution (ω) to a Laplace representation (s). Many users start from the frequency-domain representation of solutions of wave propagation problems. In this case issues arising from the movement of singularities under the transformation from ω to s must be considered. We discuss this extension in the context of the Sommerfeld half-plane problem.