Samuel Leonard Butler

• PhD
• University of Saskatchewan at University of Saskatchewan

Signal contribution functions can be integrated over space to calculate the response of an electrical or electromagnetic technique to a given resistivity distribution. On the other hand, sensitivity functions show how the measured signal changes with a change in resistivity in a region of the ground. Signal contribution functions and sensitivity ha...

In the electrical resistivity method, electrodes are usually modelled as point current sources and point voltage measurements. If the burial depth of the electrode is significant compared with the spacing between electrodes, this point approximation may not be accurate. Common situations employing long electrodes include the use of metal‐cased bore...

Computing the velocity field is an expensive process for mantle convection codes. This has implications for particle methods used to model the advection of quantities such as temperature or composition. A common choice for the numerical treatment of particle trajectories is classical fourth‐order explicit Runge–Kutta (ERK4) integration, which invol...

Equations describing mushy systems, in which solid and liquid are described as a single continuum, have been extensively studied. Most studies of mushy layers have assumed them to be ‘ideal’, such that the liquid and solid were in perfect thermodynamic equilibrium. It has become possible to simulate flows of passive porous media at the pore scale,...

Computing the velocity field is an expensive process for mantle convection codes. This has implications for particle methods used to model the advection of quantities such as temperature or composition. A common choice for the numerical treatment of particle trajectories is classical fourth-order Runge–Kutta (ERK4) integration, which involves a vel...

Within density functional theory, we have studied self-sustained, deformable, rotating liquid He cylinders subject to planar deformations. In the normal fluid He3 case, the kinetic energy has been incorporated in a semiclassical Thomas-Fermi approximation. In the He4 case, our approach takes into account its superfluid character. For this study, we...

In this study, we manufacture an exact solution for a set of 2D thermochemical mantle convection problems. The derivation begins with the specification of a stream function corresponding to a non‐stationary velocity field. The method of characteristics is then applied to determine an expression for composition consistent with the velocity field. Th...

We have studied self-sustained, deformable, rotating liquid He cylinders of infinite length. In the normal fluid $^3$He case, we have employed a classical model where only surface tension and centrifugal forces are taken into account, as well as the Density Functional Theory (DFT) approach in conjunction with a semi-classical Thomas-Fermi approxima...

The shapes of rotating fluid drops held together by surface tension is an important field of study in fluid mechanics. Recently, experiments with micron-scale droplets of liquid helium have been undertaken and it has proven useful to compare the shapes of the resultant superfluid droplets with classical analogs. If the helium is a mixture of He3 an...

The electrical conductivity method in boreholes has been applied for exploration as well as engineering and environmental investigations. The simplest deployment involves placing electrodes at varying heights within a single borehole. Borehole surveys differ from surface surveys using colinear arrays in that the ground surface is in the line of the...

Magnetotellurics is an electromagnetic geophysical method that has been widely used to study structures in Earth's subsurface. Numerical modeling of magnetotellurics is important for survey design, inversion, geological interpretation and many other aspects of geophysical studies. For example, modeling a subsurface conductive body in terms of its c...

Numerical simulations are a highly valuable tool for improving our understanding of mantle dynamics. COMSOL Multiphysics® is a commercial software suite designed to numerically model experiments featuring multiple branches of physics. This modeling approach applies to mantle convection, which can be viewed as a combination of fluid dynamics and the...

We analyze a two-phase porous medium whose permeability and solid viscosity are dependent on porosity. It has been established experimentally and numerically that when such a medium is subjected to shear, the porosity rearranges into stripes of high and low porosity known as melt bands (Holtzman et al., 2003; Katz et al., 2006; Butler, 2009). This...

Modeling the evolution of composition in a convecting mantle is difficult since the associated chemical diffusivity is very small. Consequently, compositional evolution is often modeled using the advection equation which is prone to overdiffusion and spurious oscillations unless special numerical schemes are employed. Similar errors can also occur...

While rotating single phase fluid drops have been thoroughly investigated, the determination of the shape of a drop consisting of two immiscible liquid phases has not been previously considered. Recently, experiments using rotating micron-scale droplets of liquid helium have been carried out where the liquid can be in a normal or superfluid state d...

In the geophysical electrical resistivity method, arrays of electrodes are commonly deployed on the Earth's surface. In electrical power engineering applications, arrays of electrodes are often connected in parallel in order to reduce the total resistance to ground. The electrical characteristics of an array of N electrodes can be fully described b...

It is instructive to consider the sensitivity function for a homogeneous half space for resistivity since it has a simple mathematical formula and it does not require a priori knowledge of the resistivity of the ground. Past analyses of this function have allowed visualization of the regions that contribute most to apparent resistivity measurements...

Rotating superfluid He droplets of approximately 1 μm in diameter were obtained in a free nozzle beam expansion of liquid He in vacuum and were studied by single-shot coherent diffractive imaging using an x-ray free electron laser. The formation of strongly deformed droplets is evidenced by large anisotropies and intensity anomalies (streaks) in th...

The applied potential, or mise-à-la-masse, method is used in mineral exploration and environmental applications to constrain the shape and extent of conductive anomalies. However, few simple calculations exist to help gain understanding and intuition regarding the pattern of measured electrical potential at the ground surface. While it makes intuit...

Shear-induced porosity bands have been observed experimentally and have been the subject of a number of theoretical and numerical analyses in which a number of rheological laws governing the partial melt system have been proposed. These bands have been suggested to be important in Earth’s interior in focussing melt to Earth’s mid-ocean ridges, in r...

Until now, a simple formula to estimate the depth of investigation of the electrical resistivity method that takes into account the positions of all of the electrodes for a general four-electrode array has not been available. While the depth sensitivity function of the method for a homogeneous infinite half-space is well known, previous attempts to...

Imposing external shear on systems of partial melt can cause compaction of the solid matrix and concentration of the interstitial liquid melt thereby generating bands of contrasting high and low porosity. These shear-induced porosity bands have been proposed to channel melt beneath a mid-ocean ridge (MOR). In this contribution, we impose a shear fl...

In geophysical electromagnetic methods, time-varying magnetic fields are measured at Earth's surface that are produced by electrical currents inside the Earth in order to constrain subsurface conductivity and geological structure. These methods are widely used for mineral exploration and environmental investigations, and are increasingly being used...

Mantle convection models with a low viscosity asthenosphere and high viscosity surface plates have been shown to produce very large aspect ratio convection cells like those inferred to exist in Earth's mantle and to exhibit two asthenospheric flow regimes. When the surface plate is highly mobile, the plate velocity exceeds the flow velocities in th...

A common error in the electrical resistivity method occurs when a cable connecting to a current or potential electrode is inadvertently grounded at a point other than the intended electrode, thus creating an extra electrode. In this paper, we derive expressions for the magnitude of the induced error of the inadvertent electrode as a function of the...

Determining the shapes of a rotating liquid droplet bound by surface tension is an archetypal problem in the study of the equilibrium shapes of a spinning and charged droplet, a problem that unites models of the stability of the atomic nucleus with the shapes of astronomical-scale, gravitationally-bound masses. The shapes of highly deformed droplet...

The use of numerical simulations to model physical processes occurring within subvolumes of rock samples that have been characterized using advanced 3D imaging techniques is becoming increasingly common. Not only do these simulations allow for the determination of macroscopic properties like hydraulic permeability and electrical formation factor, b...

Ten splash-form tektites from the Australasian strewn field, with masses ranging from 21.20 to 175.00 g and exhibiting a variety of shapes (teardrop, ellipsoid, dumbbell, disk), have been imaged using a high-resolution laser digitizer. Despite challenges due to the samples’ rounded shapes and pitted surfaces, the images were combined to create 3-D...

3-D laser imaging has been used previously to model tektites. These models can be used to find density and to calculate the inertia and rotation period.

When a system of partial melt is subjected to an externally driven strain-rate, an instability can occur whereby bands of low and high porosity form. Theory and numerical simulations have shown that if the matrix viscosity is isotropic and strain-rate independent, the bands grow fastest when parallel to the direction of maximum compression of the e...

Forward modeling is useful in geophysics both as a tool to interpret data in a research setting and as a tool to develop physical understanding in an educational setting. Gravity, magnetics, resistivity, and induced polarization are methods used in applied geophysics to probe Earth's subsurface. In this contribution, we present forward models of th...

A compacting porous medium, consisting of a high viscosity matrix with a low viscosity interstitial fluid, will undergo an instability when sheared whereby low and high porosity bands are formed. The bands may act as high permeability conduits for melt in the upper mantle. When the matrix viscosity is isotropic, linear theory and numerical simulati...

Splash-form tektites are glassy rocks ranging in size from roughly 1 to 100 mm that are believed to have formed from the splash of silicate liquid after a large terrestrial impact from which they are strewn over thousands of kilometres. They are found in an array of shapes including spheres, oblate ellipsoids, dumbbells, rods and possibly fragments...

Mushy layers are known to occur in magma chambers, sea-ice, and metal castings. They are often modeled as a porous layer in which a fluid and solid matrix exist in thermal and compositional equilibria. In nonreactive porous media, both advective and diffusive transport rates for heat and solute differ. In mushy layers, however, the temperature and...

Splash-form tektites are found with a wide range of sizes and in an intriguing array of shapes ranging from spheres to flat discs to dumbbells. Despite the considerable interest that exists in tektites, there has been relatively little effort to develop rational shape descriptors and to understand the origin of their shapes based on basic physics....

A porosity localizing instability occurs in compacting porous media that are subjected to shear if the viscosity of the solid matrix decreases with porosity (Stevenson, 1989). This instability may have significant consequences for melt transport in regions of partial melt in the mantle and may significantly modify the effective viscosity of the ast...

The convective Urey ratio is equal to the instantaneous heating generated in the Earth's mantle by radioactive decay divided by the contribution of convection in Earth's mantle to Earth's surface heat flow. The measured heat flow at the Earth's surface as well as geochemical models for radioactive abundances give relatively low modern-day convectiv...

Melt regions in the Earth's mantle can undergo compaction phenomena. When a compacting layer whose matrix weakens with porosity is strained, a porosity localizing instability occurs producing bands of high and low porosity that are parallel (for the case of strain-rate independent viscosity) or at small angles to the direction of maximum compressio...

When a compacting porous layer, modeled as two interpenetrating viscous fluids, is subjected to an external stress, it has been shown that if the viscosity of the solid matrix decreases with porosity, that an instability occurs that results in localized regions of high and low porosity. This instability was first predicted theoretically for a pure...

The thermal insulation caused by partial continental coverage induces very large aspect ratio convection cells in mantle convection models with upwellings beneath the continents. The ability of upwellings and downwellings to penetrate the 660-km depth endothermic phase boundary has been shown to increase with the wavelength of convection. In this s...

Physics of the Earth and Planetary Interiors j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / p e p i a b s t r a c t We explore a wide range of parameter space to investigate the dynamical interaction between effects due to surface boundary conditions representing continental and oceanic lithosphere and the endotherm...

It has recently been shown [Holtzman, B., Groebner, N., Zimmerman, M., Ginsberg, S., Kohlstedt, D., 2003. Stress-driven melt segregation in partially molten rocks. Geochem. Geophys. Geosyst. 4, Art. No. 8607; Holtzman, B.K., Kohlstedt, D.L., 2007. Stress-driven melt segregation and strain partitioning in partially molten rocks: effects of stress an...

Splash-form tektites are believed to represent molten rock that was ejected by a large impactor that solidified while in flight. These glassy rocks are found in a number of strewn fields around the Earth and are found in a number of intriguing shapes including near spheres, axisymmetric biconcave shapes as well as rods and "dumb-bells". In this con...

Recent seismic tomographic images indicate that slabs penetrate through the 660-km depth phase boundary in some locations while they are deflected in their trajectory in others, suggesting that convection in Earth's mantle is currently partially layered at this depth. The efficiency of heat transport by convection decreases with increasing layering...

In this contribution we present the results of numerical simulations of the thermal evolution of the Earth assuming a 200-km layer with high internal heating at the top of the core. Our models are inspired from recent isotopic studies that suggest the presence of a region with high internal heating residing in D' ' as an early enriched reservoir in...

It is well established that the temperature gradients in the interiors of internally heated mantle convection models are subadiabatic. The subadiabatic gradients have been explained as arising because of a balance between vertical advection and internal heating; however, a detailed analysis of the energy balance in the subadiabatic regions has not...

We present a suite of high-resolution numerical model experiments conducted to investigate the effects of varying thermal and compositional buoyancy on the behavior and morphology of plumes in porous media. The calculations model the injection of fluid through a narrow opening into the base of a nonreactive, saturated, porous matrix with interstiti...

Partial melting occurs beneath mid-ocean ridges in Earth’s mantle and the resulting liquid migrates to the surface to form new oceanic crust. In this system, mass can be exchanged between the liquid and solid phases through melting and solidification and, at the high temperatures and pressures associated with the Earth’s interior, the solid matrix...

Recently, it has been proposed that a layer of high internal heating exists in D". Arguments for such a layer have come both from geochemical considerations and from thermal modeling of core evolution. In this contribution, we present the results of numerical modeling of the thermal evolution of the Earth with high degrees of internal heating in th...

It has been known for some time that the geotherm calculated in models of convection in Earth's mantle is subadiabatic when internal heat sources, used to model the effects of radioactive decay, are present. It has been argued that such subadiabaticity arises because of a balance in the energy equation between vertical advection and internal heatin...

Recently, an incompatible-element enriched reservoir, bearing a high degree of radioactive heating, has been proposed to exist at the base of the mantle. This scenario has been discussed based on parameterized thermal and magnetic models of the core [Buffett, B.A., 2002. Estimates of heat flow in the deep mantle based on the power requirements for...

We describe a series of numerical simulations of dissolution-driven convection in a reactive porous medium heated from above. The physical system consists of a porous medium made of the frozen component of a binary mixture that is immersed in a liquid mixture with which it is in thermodynamic equilibrium. Surface heating results in melting of the u...

A reactive porous medium is one in which an interstitial fluid may exchange mass through melting and freezing with a solid matrix. In a multi-component system, fractionation may occur which can have a significant impact on the density of the interstitial fluid which can, in turn, lead to compositionally driven convection within the porous layer. Th...

Recently there has been renewed interest in the evolution of the inner core and in the possibility that radioactive potassium might be found in significant quantities in the core. The arguments for core potassium come from considerations of the age of the inner core and the energy required to sustain the geodynamo [Nimmo, F., Price, G.D., Brodholt,...

A number of different scenarios have emerged for the thermal evolution of the Earth which is controlled by the slow convective motion in the mantle in order that it meet the constraint imposed by the modern-day surface heat flow. These include models with high internal heating rates in the mantle, models with significant degrees of internal heating...

Although the primary impact of the mantle convection process, apart from its role as prime mover of surface plate tectonics, is through the control it exerts upon the rate of radial heat transport, little or no use is normally made of the constraint upon convection models provided by surface heat flow measurements. In this paper we will both review...

We employ an axisymmetric spherical shell model of mantle convection to examine the magnitude of deviatoric tensile stresses generated in a stationary continental plate resulting from the subduction of oceanic plate material below an active continental margin. The model includes depth-dependent physical properties, uniform internal heating, compres...

The amount of ocean floor topography caused by upwelling mantle plumes can be used as a constraint on the degree of layering of convection in the Earth's mantle (Davies, 1998). In this contribution, we present a suite of numerical models of convection in the Earth's mantle with varying degrees of layering at 660 km depth. An oceanic plate is modele...

Recent studies of energy and entropy balances in the core constrain the power requirements for magnetic field generation and the age of the inner core (\textit{e.g.} Buffett, 2002, Labrosse et al., 2001, Labrosse, 2003, Nimmo et al., 2004). These studies suggest that the change of entropy in the core needs to be positive during the entire period th...

Due to the multi-component nature of magmas, cooling of magma chambers may lead to the formation of mushy layers in the thermal boundary layers adjacent to the host rocks. Further differentiation may take place if the fluid within the mushy layer becomes convectively unstable. Analog laboratory experiments using aqueous solutions have provided sign...

Thermal convection in the Earth's mantle regulates the rate at which heat is lost from the Earth's interior. Factors affecting the vigor of that convection, such as the viscosity of the mantle and barriers to convection such as the 660-km endothermic phase transition, as well as the rate of radioactive heat generation within the planet determine th...

We present a new set of Earth thermal history calculations in which the effect of increasing mantle layering with convection Rayleigh number is included in a parameterized mantle convection model. We demonstrate that the inclusion of this effect results in strong buffering of the upper mantle temperature and surface heat flow. Models of this type d...

The question of the mechanism that is responsible for super-continent break-up has been a long-standing one. Active mantle plume upwellings, down-going oceanic plates at continental margins, and passive upwellings due to continental insulation of the mantle have all been proposed as mechanisms capable of creating sufficient extensional stresses wit...

Thesis (Ph.D.)--University of Toronto, 2001. Includes bibliographical references.

We present a series of simulations of the mantle convection process based upon an axisymmetric numerical model and highlight a wide range of results in which scaling emerges. For the more challenging simulations it was found necessary to employ a finite difference mesh with uneven grid spacing in the radial coordinate, and we present the appropriat...

The stability of a horizontal thermal boundary layer embedded within a very viscous fluid is investigated using the formalism of linear stability analysis. Thin thermal boundary layers in deep fluid regions and in the absence of phase transition and dynamical effects are thereby shown to be unstable at extremely long wavelengths. The stability of t...

As is well known, a wide variety of forced dissipative systems achieve a statistical equilibrium through restorative events which exhibit scale invariance, evidenced by power law distributions of both amplitude and duration with frequency. Geophysical examples include the distribution of earthquake seismic moments in a given region, the well known...