Edward W. Bolton
Yale University
Department of Geology and Geophysics
Please see my recent publication list in my CV for more recent research.
The natural world exhibits complexity at scales ranging from mineral grains to galactic structures. Many systems involve fluid flows, typically governed by nonlinear processes. I have taken the basic philosophy that the deepest insight into the dynamics of fluids can be gained by following flow regimes through their initial bifurcations. Analytical, numerical, and experimental work complement each other well in this approach. For thermal convection, as the control parameter (Rayleigh number) increases, the fluid layer goes through a sequence of bifurcations: from a static layer, to steady rolls, to time dependence via various instabilities. Stability analysis plays an important role in understanding these events. In addition to indicating the region of parameter space at which a solution branch may become unstable, an analysis of stability may provide the geometric form of the instability in the associated eigenvector.
Fluid and granular flows may be classified by their geometric forms. Some flows exhibit a preferred scale (e.g. convection rolls, sand ripples, beach sand cusps), while others exhibit scale independence (e.g. turbulent flows, clouds, land surfaces, each with fractal or multifractal characteristics). Flows with a preferred scale may be understood by following the initial bifurcations. Even chaotic flows retain structures remnant from primary instabilities. Turbulent flows are also strongly influenced by spatially inhomogeneous forcing. As an example, atmospheric flow responds to the strong equatorial solar forcing and inhomogeneous land/sea distribution to yield temporal and spatial means dominated by low wavenumber structure.
In my research, I concentrate on nonlinear flows and their bifurcations. Recently, I have expanded my research interests into the domain of fluid flow in the Earth's crust, modeling kinetically governed precipitation and dissolution of minerals for porous media flows in heterogeneous permeability fields, along with chemically reacting species. I have also begun modeling landform evolution via hydraulic erosion.
Please see my recent publication list in my CV for more recent research.
I have developed a two-dimensional, time-dependent code to model flow and chemical reactions in a fractured and porous medium. Both equilibrium and nonequilibrium reactions are modeled using real thermodynamic data. I am examining the effect of reaction kinetics on changes of porosity with time. Working with Antonio Lasaga and Danny Rye, we have discovered that oscillatory thermal boundary instabilities have a profound influence on the saturation state and precipitation history. Zonation of mineral grains can often be expected down to the smallest observable scale. Significant portions of crustal fluids violate assumptions based on equilibrium when real kinetics are incorporated into models. For example, near the ends of high permeability zones, one can observe downwelling oversaturated fluids and oversaturated fluids moving toward higher temperatures, both contrary to the conventional wisdom based on equilibrium. Potential applications for such models are broad, and encompass a variety of geological problems, from ore deposits, to nuclear waste contamination, to deposits in hydrothermal zones. Ultimately, we hope to couple this code to a physical erosion code (discussed below).
Working in collaboration with Mike Oristaglio's group at Schlumberger-Doll Research in Ridgefield, Connecticut, I have developed a Lagrangian particle tracking scheme for solving the advection of gas/oil/water interfaces associated with secondary oil recovery. The interface motion induces measurable changes in the gravity field. We hope to compare the results of such numerical simulations with field observation of microgravity changes in horizontal wells.
We present the formulation and model results of kinetically controlled quartz dissolution and precipitation in a two-dimensional heterogeneous permeable medium. The quartz matrix is modeled as a partially occluded spherical close pack, with dissolution and precipitation occurring on the exposed faces of the grains. This formulation yields larger permeabilities and lower surface area to fluid volume ratios in regions of larger grain radii, and thus allows us to investigate the influence of crack-like regions on the flow and silica exchange. We use the kinetic data for quartz dissolution of Rimstidt and Barnes (1980). Thermal convection results indicate that channelizing of fluid flow in high permeability zones is enhanced in the transient regime by buoyancy effects arising from the advection of heat. The highly permeable zones are most out of chemical equilibrium, owing to their lower surface areas and to more rapid advection. Porosity changes are most pronounced in regions of high surface area often downstream of high permeability zones. Oscillatory convection is observed, accompanied by saturation state reversals peripheral to the high permeability zones. Such internally generated oscillatory regimes provide a mechanism for quartz zonation down to the nanometer scale. When the thermal forcing is strong enough, recurrent plumes emanate from the thermal boundary layers and plunge through the high permeability zones. When departures from equilibrium become significant, we observe some regions of undersaturated upwelling fluid moving down temperature, and regions of oversaturated downwelling fluid moving up temperature, both cases opposing the conventional wisdom based on equilibrium.
We model kinetically controlled dissolution and precipitation of quartz in a porous medium. Our specific focus is on the spatial pattern of flow velocities and of changes in solute concentration and porosity introduced by heterogeneities in the initial permeability field. Upon a background permeability field, we impose isolated "crack-like" zones of high permeability and low surface area to fluid volume ratios. Our two-dimensional modeling of forced flux injection of over-- and under--saturated fluid reveals features inaccessible to previous homogeneous permeability studies. Although realistic velocities give rise to narrow boundary layers in the solute concentration, disequilibrium is favored in the high permeability zones yielding plume--like structures of solute concentration. Aside from the rapid changes in porosity near the injection level, the other regions of rapid porosity change occur just downstream from the cracklike zones, where fluid more out of equilibrium encounters regions of higher surface area. Flow rates are significantly enhanced even between isolated high permeability zones, an effect which is even more dramatic for both closer "crack" spacing and higher permeability contrasts. Undersaturated injection leads toward permeability homogenization along the flow direction, whereas oversaturated injection tends to increase permeability heterogeneities along the flow direction.
The long-term feedback between flow and chemistry, where dissolution and precipitation is under kinetic control, is examined for the quartz matrix system. Examples of thermal convection in a porous medium with spatially variable permeability reveal features of central importance to water-rock interaction. Kinetic effects produce features not expected by traditional assumptions made on the basis of equilibrium, e.g. that cooling fluids are oversaturated, and heating fluids are undersaturated with respect to silicic acid equilibrium. Indeed, we observe regions of downwelling oversaturated fluid experiencing heating, and upwelling, yet cooling, undersaturated fluid. In sloping high permeability zones, upwelling leads to increasing slopes of the flow with time, due to the deposition along the upper surface of the channel. In the long term, this may also lead to the onset of oscillatory behavior near the surface. Downwelling in sloping high permeability zones tends to become more vertical with time, due to buoyancy effects and dissolution at the core of the downwelling. The location of the basal stalk of thermal plumes rising from the heated lower boundary is inherently unstable. This stalk migrates with time, as the core of the flow generally clogs via precipitation, whereas the edges of the stalk are dissolving, via kinetic effects. When oscillatory convection is present, the amplitudes of oscillation generally increase with time in near-surface environments, whereas amplitudes tend to decrease over long times near the heated lower boundary. Runaway dissolution is moderated by shifts in the locations of saturation state reversals.
Land surface material is transported overland by both flowing water and gravity. The combination of tectonic and erosive forces creates the observed topography. We have incorporated simple erosion laws into a numerical finite-difference model which simulates the appearance of typical badlands topography. We solve the shallow-water equations for flow down arbitrary topography, along with an equation for mass continuity. Sediment fluxes are calculated from the water slope, depth, and velocity. The approach is more fundamental than the water-passing models which have up to now dominated the literature in dynamic geomorphology. Initial results of this work, with computer scientist Kenton Musgrave, appear realistic. Water and sediment masses are conserved; and erosion, transport, and deposition are calculated. Quantitative comparison of landform types with model results is one goal of this work, but will require an extension of wavelet analysis in order to allow basis functions of nonzero mean. This alternative to Fourier methods should have important applications to pattern recognition, especially when the structure to be analyzed is composed of a superposition of a particular geometric form at many different scales. In a complementary study, with graduate student Wenjie Zhao, we investigated the onset of the channeling instability. Flow down a sloping land surface creates channels when the hydraulic erosive effects dominate the diffusive effects.
Working with K. A. Maasch and J. M. Lilly, I recently implemented a continuous wavelet analysis of the Earth's climate record of the last 2 million years. The local nature of the wavelet basis functions enables the computation of a transform in which the time dependence of modal amplitudes is clearly apparent. I created a new normalization of the wavelet bases which allows "period" to have a meaning consistent with that of Fourier analysis. The comparison of insolation forcing with oxygen isotopic variations in sediment cores shows that both series exhibit a beating pattern of the 20 kyr periodicity. In addition, the dominance of the ~100 kyr periodicity during the last million years, is seen to enter in the records with a beating pattern of successively longer periods. I am collaborating with Mike Mann, Jonathan Lees, and Kirk Maasch on a paper in which we compare wavelet analysis with evolutive and envelope multitaper analyses to highlight their relative advantages and disadvantages when applied to nonstationary processes.
Bolton, E.W., K.A. Maasch and J. M. Lilly (1995), A wavelet analysis of Plio-Pleistocene climate indicators: A new view of periodicity evolution, Geophysical Research Letters, 22, 2753-2756
Convection in rotating and nonrotating spherical shells is of fundamental importance to outer-core convection and mantle convection, respectively. Mantle convection creates a lower thermal boundary layer which is spatially inhomogeneous. By continuity of heat flux, less heat is extracted from the outer core near upwelling mantle plumes than from regions where the mantle isotherms are compressed by the downwelling of relatively cool material. This creates a spatially inhomogeneous thermal boundary condition for outer-core convection. I have constructed numerical and experimental models to investigate outer core convection with thermal coupling at the outer boundary. The numerical (spectral) code is similar to the one I developed in my dissertation research, except that I allow full time-dependence. I have taken considerable pains to optimize the calculation of the nonlinear terms. By pre-summing repeated factors, I believe the code is competitive with mixed spectral/finite difference codes. Although it would be most relevant to impose spatially varying heat flux conditions, this turns out to be a difficult problem. Instead, I have imposed a temperature boundary condition. Although the core/mantle boundary is generally thought to be isothermal, the overall effect on core convection should be similar, whether the variable condition is for temperature or heat flux. The temperature at the outer spherical surface (core/mantle boundary) is specified as a constant plus a factor, E, times a low order spherical harmonic. Because the temperature varies laterally, fluid motion is forced. If E is large enough, the normally drifting thermal Rossby waves can become locked onto the inhomogeneity [Bolton and Sayler 1990, abst.].
I developed a code for temperature-dependent viscosity convection in a spherical shell, at infinite Prandtl number. Nonlinear model simulation results were used to calculate the expected anisotropy using a theory developed by Neil Ribe [Bolton and Park, 1991, abst.].
I also developed a code for convection and magnetic field generation in rotating, self-gravitating, spherical shells [Bolton, Ph.D. dissertation 1985]. This study was motivated by the problem of the geodynamo, and magnetic field generation in the major planets. The velocity, magnetic, and thermal fields are expanded in terms of spherical harmonics, and radial functions which satisfy the boundary conditions. We considered only traveling wave solutions. We were successful in finding nonlinear convection solutions over a wide range of parameter space, but solutions with magnetic fields eluded us.
Ph.D. DISSERTATION: Problems in Nonlinear Convection in Planar and Spherical Geometries, University of California, Los Angeles, (Advisor: F.H. Busse) 1985
E.W. Bolton and F.H. Busse, Nonlinear thermal convection in rotating spherical shells, paper presented on 4 Aug. 87 at the Nonlinear Dynamics of Rotating Magnetic Systems Conference at University of California, Los Angeles, 1987.
E.W. Bolton and F.H. Busse, Nonlinear thermal convection in rotating spherical shells, paper GA1.3-7 presented on 19 Aug. 87 at the International Union of Geodesy and Geophysics (IUGG) at Vancouver, Canada, 1987.
E.W. Bolton and B. Sayler, The influence of lateral variations of thermal boundary conditions on core convection: Numerical and laboratory experiments, presented at Santa Fe, Studies of the Earth's Deep Interior (SEDI) symposium, August, 1990.
E.W. Bolton and J. Park, The development of anisotropy in a convecting mantle, EOS, Trans of Am. Geoph. Union, 72, p. 508 of Fall 1991 AGU Meeting Supplement, 1991.
I have developed a simple notation for differential vector operations in orthogonal curvilinear coordinates, with which I derive a compact formula for the divergence of general second-order tensors. This notation is not limited to Euclidean space, and therefore provides an efficient alternative to the use of covariant and contravariant vectors and the Christoffel symbols.
As previous Galerkin codes for spherical shell convection were based on nonorthogonal radial expansion functions, I have developed an orthogonal radial basis set for the poloidal velocity field, which satisfies two boundary conditions at each spherical surface. For the perturbation temperature and toroidal velocity fields, a combination of spherical Bessel functions of the first and second kind yield a radial basis set. I have also derived a family of orthogonal polynomial basis functions which appear to better resolve the radial structure [Bolton, 1992, abst.].
E.W. Bolton, Radial functions for spectral/Galerkin modeling of spherical-shell convection, EOS, Trans of Am. Geoph. Union, 73, p. 575 of Fall 1992 AGU Meeting Supplement, 1992.
I worked with Neil Ribe on two problems relevant to upper mantle "fluid" dynamics. I developed a numerical code to examine entrainment and mixing in an infinite Prandtl number fluid with temperature- and concentration-dependent viscosity. The flow is driven by temperature gradients, while being partially stabilized by concentration gradients. This work is being compared with recent experimental high Rayleigh number convection studies of Anne Davaille. In another project with Ribe, we examined the lubrication limit behavior of a large viscous drop (or "splat") in contact with a rigid upper lid. The splat was assumed to be embedded in a background fluid possessing a constant shear. I developed a numerical code for the evolution of such a splat, as a model for thermal plumes interacting with the Earth's lithosphere.
Nonlinear convection in plane layers may take on a variety of planforms when the viscosity is temperature dependent. For the square planform case, we have developed neutral curves and small-amplitude results, as well as a fully nonlinear code [Bolton and Ribe, 1989, abst.].
E.W. Bolton and N.M. Ribe, Square-cell convection in a fluid with temperature dependent viscosity, EOS, Trans of Am. Geoph. Union, 70, p. 1333, 1989.
Convection driven by centrifugal buoyancy in a rotating annulus with sloping end boundaries creates drifting thermal-Rossby waves. The convection columns align with the rotational axis. This flow regime is dynamically similar to convection in the Earth's liquid core and in the deep atmospheres of the major planets. In an experimental study [Azouni, Bolton, and Busse 1986], we were the first to collect time series and convective amplitudes of the traveling waves induced by sloping end boundaries (such waves had been previously measured in wide gap annulus experiments with non-sloping end boundaries by White and Koschmieder, 1981; Koschmieder and White, 1981). The convective amplitudes were found to reach a maximum and then decay as the Rayleigh number was increased.
I have constructed a rotating annulus convection experiment with periodic heating wires in one of the cylindrical cell walls [Bolton and Sayler, 1990, abst.]. The heating wires are parallel to the rotational axis and are spaced at the wavelength expected for the onset of convection. In addition to the periodic heating, the cell walls are adjacent to uniform warm temperatures on the outside, and cooling is provided on the cell interior. Sloping end boundaries force thermal Rossby waves, which drift in the prograde sense. By increasing the ratio of inhomogeneous to homogeneous heating, I expect to observe the locking phenomena. Up to now, the results with the heating wires turned on have not been conclusive, due to excessive noise. I am redesigning the means to supply power to the heating wires. I set up a data acquisition system using National Instrument hardware and software in a Macintosh IIx computer. With only homogeneous heating, the results are of much better quality than Azouni, Bolton, and Busse [1986].
Azouni, M.A., E.W. Bolton and F.H. Busse, Convection driven by centrifugal buoyancy in a rotating annulus, Geophys. Astroph. Fluid Dyn., 34, 301-317, 1986.
I.A. Quintanar, E.W. Bolton, F.H. Busse and M.A. Azouni, Experimental measurements of drifting convection columns in a rotating annulus, paper presented on 6 Dec. 85 at the Fall Meeting of the Am. Geoph. Union, San Francisco, CA, EOS, Trans. Am. Geoph. Union, 65, No. 45, p. 871, 1984.
E.W. Bolton and B. Sayler, The influence of lateral variations of thermal boundary conditions on core convection: Numerical and laboratory experiments, presented at Santa Fe, Studies of the Earth's Deep Interior (SEDI) symposium, August, 1990.
Bubbles rising in a water-filled inclined cell, with a gap spacing of about 4 mm, will generate a vortex street. In a series of experiments with H.F. Bolton, we characterized the terminal velocities, neutral curve, and amplitude and frequency of oscillations observed [Bolton and Bolton, 1988, abst.]. Most studies of periodically forced fluid flows impose the frequency and the amplitude of the forcing. What happens when the forcing frequency is itself created by the flow? We created a cylindrical annulus 20 cm long, with a gap spacing of 2.4 mm and a mean gap diameter of 1.3 cm. The cell was filled with water. A single bubble of air (with a diameter between 4 and 12 mm) was situated in the center of the length, while the axis was horizontal. Upon rotation of this cell, the bubble oscillates from side to side. The generated vortex street rotates with the cylinder, and then forces the bubble at the same frequency as its previous oscillation. Video techniques were used for data acquisition, and the bubble position was digitized to yield a time series. Fourier analysis indicated several regimes (including simply periodic, doubly periodic, and chaotic), in the parameter space of bubble diameter and rotation rate. A simple numerical model of a Hopf bifurcation with delayed feed-back qualitatively yields some of the same results as the experiment [Bolton and Sayler, 1989, abst.].
E.W. Bolton and H.F. Bolton, Vortex street generation and bubble motion in inclined fluid planes, presented at the Division of Fluid Dynamics Meeting of The American Physical Society at Buffalo, NY, Bulletin of the American Physical Society, 33, No. 10, p. 2253, 1988.
E.W. Bolton and B. Sayler, Karman vortex feedback and chaotic bubble motion in a rotating annulus, presented at the Division of Fluid Dynamics Meeting of The American Physical Society at Palo Alto, CA, Bulletin of the American Physical Society, 34, No. 10, p. 2331, 1989.
Thermal convection in horizontal plane layers is widely regarded as a classic example of hydrodynamic instability. In the companion papers Busse and Bolton [1984] and Bolton and Busse [1985], we examined the instabilities of nonlinear convection rolls with stress-free boundary conditions. Both the numerical and analytical studies indicated that, except at infinite Prandtl number, the convection rolls with the critical wavenumber are unstable to the skewed-varicose instability. We used a perturbation expansion technique for the analytical paper, and a Galerkin technique for the numerical paper.
In another paper [Fauve, Bolton, and Brachet 1987], we developed phase equations for the oscillatory instability of both rigid and free-slip horizontal-layer convection. This was the first paper in which the coefficients of phase equations were calculated from nonlinear simulations. We found that the oscillatory instability should consist of traveling waves, rather than standing waves. In addition, we predicted that, for convection of mercury, the oscillatory instability should have a subcritical onset for low wavenumbers.
In a second numerical study [Bolton, Busse, and Clever 1986], we examined the stability of convection rolls with rigid boundary conditions at intermediate Prandtl numbers. A family of oscillatory "blob" instabilities were discovered to limit the region of stable rolls at high Rayleigh numbers and low wavenumbers. We were also successful in characterizing the transition from the knot to the cross-roll instability.
Nonlinear convection in plane layers may take on a variety of planforms when the viscosity is temperature dependent. For the square planform case, we have developed neutral curves and small-amplitude results, as well as a fully nonlinear code [Bolton and Ribe, 1989, abst.].
Busse, F.H. and E.W. Bolton (1984) Instabilities of convection rolls with stress-free boundaries near threshold, J. Fluid Mech., 146, 115-125.
Bolton, E.W. and F.H. Busse (1985) Stability of convection rolls in a layer with stress-free boundaries, J. Fluid Mech., 150, 487-498.
Bolton, E.W., F.H. Busse and R.M. Clever (1986) Oscillatory instabilities of convection rolls at intermediate Prandtl numbers, J. Fluid Mech., 164, 469-485.
Fauve, S., E.W. Bolton and M.E. Brachet (1987) Nonlinear oscillatory convection: A quantitative phase dynamics approach, Physica, 29D, 202-214.
E.W. Bolton and N.M. Ribe (1989) Square-cell convection in a fluid with temperature dependent viscosity, EOS, Trans of Am. Geoph. Union, 70, p. 1333.
I performed a series of experiments on a new instability of an oscillatory shear layer with Jean Maurer. In addition to developing an analytical description of the time-dependent basic state, I developed a resonance model which shows some promise for understanding the bifurcation to strong rolls at high frequencies [Bolton and Maurer 1994].
Bolton, E.W. and J. Maurer (1994) A new roll-type instability in an oscillating fluid plane, J. Fluid Mech, 268, 293-313.
Visualization of fluid motion is of central importance to the development of both intuition and theory. To this end I have developed devices which illustrate the following instabilities:
Kinetic art sculptures which I designed or made design contributions have been on display at two exhibitions of Leonardo da Vinci (The American Museum of Natural History and The Eli Whitney Museum) A 10 foot high bubble/flow sculpture was on dispay at a local bar/restaurant (BAR, 254 Crown Street New Haven, CT) from 28 September 1996 through March 1997.
I hope to continue work on numerical modeling of flow, solute transport, and reactions in a porous medium, related to fluid flows in the Earth's crust. In addition to the quartz dissolution study already completed, I have included other minerals (e.g. albite, clays, and carbonates), and model isotopic and trace element exchange between the solid and fluid phases. I also plan to continue work on erosional instabilities. This numerical approach to geomorphic landform evolution will help constrain empirical erosion laws and eventually allow quantitative predictions for erosion under flood conditions. Coupling of the surface and groundwater flows will allow quantitative modeling of the carbon cycle over geologic time.
Last updated: 11 August 2005