Bruce J Bayly
 Associate Professor, Mathematics
Contact
 (520) 6214766
 Environment and Natural Res. 2, Rm. N203
 Tucson, AZ 85719
 bjb@math.arizona.edu
Degrees
 Ph.D. Applied Mathematics
 Princeton University, Princeton, New Jersey, USA
 Kinematic and Dynamic problems in Three Dimensional Steady Flows
 B.A. Mathematics
 Cambridge University, Cambridge, Britain
Work Experience
 University of Arizona, Tucson, Arizona (1987  Ongoing)
 Courant Institute, New York University (1986  1988)
Awards
 Henry and Phyllis Koffler Prize in Public Service / Outreach
 University of Arizona, Spring 2014
Interests
Teaching
Differential equations for science and engineering majors. Applied linear algebra, matrix analysis. Advanced mathematical methods for science and engineering majors. Mathematical modeling.
Research
Fluid mechanics. Magnetohydrodynamics, Magnetic field generation in stars and planets. Math education, informal education. Math and science outreach.
Courses
202021 Courses

Intro Ord Diff Equations
MATH 254 (Fall 2020)
201920 Courses

Adv Applied Mathematics
MATH 422 (Summer I 2020) 
Adv Applied Mathematics
MATH 522 (Summer I 2020) 
Honors Thesis
MATH 498H (Spring 2020) 
Special Tops in Social Science
HNRS 195H (Spring 2020) 
Calc Concepts: Business
MATH 116 (Fall 2019) 
Honors Thesis
MATH 498H (Fall 2019) 
Intro Ord Diff Equations
MATH 254 (Fall 2019)
201819 Courses

Adv Applied Mathematics
MATH 422 (Summer I 2019) 
Adv Applied Mathematics
MATH 522 (Summer I 2019) 
Financial Math
DATA 462 (Summer I 2019) 
Financial Math
MATH 462 (Summer I 2019) 
Calculus II
MATH 129 (Spring 2019) 
Financial Math
DATA 462 (Spring 2019) 
Financial Math
MATH 462 (Spring 2019) 
Special Topics in Science
HNRS 195I (Spring 2019) 
Intro Ord Diff Equations
MATH 254 (Fall 2018)
201718 Courses

Adv Applied Mathematics
MATH 422 (Summer I 2018) 
Adv Applied Mathematics
MATH 522 (Summer I 2018) 
Calc Concepts: Business
MATH 116 (Spring 2018) 
Financial Math
MATH 462 (Spring 2018) 
Special Topics in Science
HNRS 195I (Spring 2018)
201617 Courses

Advncd Tpcs in Undergrad Math
MATH 496T (Spring 2017) 
Math Analysis Engineers
MATH 322 (Spring 2017) 
Applied Linear Algebra
MATH 310 (Fall 2016) 
Math In Modern Society
MATH 105 (Fall 2016) 
Special Topics in Science
HNRS 195I (Fall 2016)
201516 Courses

Applied Linear Algebra
MATH 310 (Summer I 2016) 
Independent Study
MATH 499 (Summer I 2016) 
Intro Math of Investing
MATH 368 (Summer I 2016) 
Intro Ord Diff Equations
MATH 254 (Summer I 2016) 
Independent Study
MATH 499 (Spring 2016) 
Intro Math of Investing
MATH 368 (Spring 2016) 
Intro Ord Diff Equations
MATH 254 (Spring 2016)
Scholarly Contributions
Journals/Publications
 Bazant, M. Z., Chu, K. T., & Bayly, B. J. (2005). Currentvoltage relations for electrochemical thin films. SIAM Journal on Applied Mathematics, 65(5), 14631484.More infoAbstract: The DC response of an electrochemical thin film, such as the separator in a microbattery, is analyzed by solving the PoissonNernstPlanck equations, subject to boundary conditions appropriate for an electrolytic/galvanic cell. The model system consists of a binary electrolyte between parallelplate electrodes, each possessing a compact Stern layer, which mediates Faradaic reactions with nonlinear ButlerVolmer kinetics. Analytical results are obtained by matched asymptotic expansions in the limit of thin double layers and compared with full numerical solutions. The analysis shows that (i) decreasing the system size relative to the Debye screening length decreases the voltage of the cell and allows currents higher than the classical diffusionlimited current; (ii) finite reaction rates lead to the important possibility of a reactionlimited current; (iii) the Sternlayer capacitance is critical for allowing the cell to achieve currents above the reactionlimited current; and (iv) all polarographic (currentvoltage) curves tend to the same limit as reaction kinetics become fast. Dimensional analysis, however, shows that "fast" reactions tend to become "slow" with decreasing system size, so the nonlinear effects of surface polarization may dominate the DC response of thin films. © 2005 Society for Industrial and Applied Mathematics.
 Bayly, B. J. (2004). Asymptotic structure of fast dynamo eigenfunctions. Fluid Mechanics and its Applications, 71, 157168.More infoAbstract: The eigenfunctions of the kinematic dynamo problem exhibit complicated spatial structure when the magnetic diffusivity is small. When the base flow is spatially periodic, we may study this structure by examining the Fourier components of the eigenfunction at large wavevectors. In this regime we may seek a WKB form in terms of slowlyvarying functions of wavevector. The resulting hierarchy of equations may be systematically analysed for both zero and small nonzero diffusivities. © 2002 Kluwer Academic Publishers.
 Bayly, B. J., Holm, D. D., & Lifschitz, A. (1996). Threedimensional stability of elliptical vortex columns in external strain flows. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 354(1709), 895926.More infoAbstract: The KirchhoffKida family of elliptical vortex columns in flows with uniform strain and rotation displays a rich variety of dynamical behaviours, even in a purely twodimensional setting. In this paper, we address the stability of these columns with respect to threedimensional perturbations via the geometrical optics method. In the case when the external strain is equal to zero, the analysis reduces to the stability of a steady elliptical vortex in a rotating frame. When the external strain is nonzero, the stability analysis reduces to the theory of a Schrödinger equation with quasiperiodic potential. We present stability results for a variety of different KirchhoffKida flows. The vortex columns are typically unstable except when the interior vorticity is approximately the negative of the background vorticity, so that the flow in the inertial frame is nearly a potential flow.
 Bayly, B. J. (1993). Scalar dynamo models. Geophysical and Astrophysical Fluid Dynamics, 73(14), 6174.More infoAbstract: The equation (δt + u•∇)C = R(x, t)C + κ∇2C, is a scalar analogue of the magnetic induction equation. If the velocity field u(x, t) and the 'stretching' function R(x, t) are explicitly given, then we have the analogue of the dynamo problem. The scalar problem displays many of the same features as the vector kinematic dynamo problem. The fastest growing modes have growth rates that approach a finite limit as K→0 while the eigenfunctions develop more and more complex structure at smaller and smaller length scales. Some insight is provided by an analysis which finds a lower bound on the growth rate that is asymptotically independent of the diffusivity © 1993 Gordon and Breach Science Publishers S.A.
 Bayly, B. J., Levermore, C. D., & Passot, T. (1992). Density variations in weakly compressible flows. Physics of Fluids A, 4(5), 945954.More infoAbstract: Density variations in real fluids are related to both pressure and entropy variations, even in the incompressible limit. The behavior of density variations depends crucially on the relative sizes of the pressure, temperature, and entropy fluctuations. It is shown how this arises from a formal asymptotic expansion of the compressible NavierStokes equations about a uniform state. Direct numerical simulations of the full compressible equations verify the consistency of different asymptotic regimes. In the case of turbulent flow, the KolmogorovObukhov theory allows one to predict inertial range scalings for the density fluctuation power spectra in various situations. The results may be relevant to observations of the interstellar medium. © 1992 American Institute of Physics.
 Cohen, J., Amitay, M., & Bayly, B. J. (1992). Laminarturbulent transition of walljet flows subjected to blowing and suction. Physics of Fluids A, 4(2), 283289.More infoAbstract: A new family of solutions describing the mean flow of an incompressible twodimensional wall jet subjected to boundary suction or blowing was found, in which the vertical velocity at the wall varies in a power law with downstream distance. Linear stability calculations indicate that two unstable modes, an inviscid inflectional point instability and a viscous instability, may coexist in the flow. It is demonstrated theoretically that the relative importance of each mode can be controlled by subjecting the wall jet to small amounts of blowing or suction. Experimental results, in the absence of suction and blowing, support theoretical findings about the coexistence of the two instability modes. © 1992 American Institute of Physics.
 Bayly, B. J., & Herbert, T. (1988). Instability mechanisms in shearflow transition.. ANN. REV. FLUID MECH., 20 , 1988, 359391.More infoAbstract: Results from classical inviscid and viscous stability theory are reviewed in relation to turbulent shear flow at high Reynolds numbers. The nonlinear growth and equilibration of two dimensional waves in wall bounded and free shear flows is next considered. Three dimensional secondary instability is then described and interpreted in terms of the theory of instability of elliptical vortices.
 Bayly, B. J. (1987). THREEDIMENSIONAL INSTABILITIES IN QUASITWO DIMENSIONAL INVISCID FLOWS.. American Society of Mechanical Engineers, Applied Mechanics Division, AMD, 87, 7177.More infoAbstract: The threedimensional inviscid NavierStokes equations have a large family of exact steady 'quasitwodimensional' solutions, in which the velocity in the x,y plane is determined by a stream function psi (x,y), with the zvelocity and zvorticity functions of psi (x,y) alone. If the projected streamlines in the x,y plane are closed curves, the flow may be subject to a broad band threedimensional instability in the form of a wave packet centered on a particular surface of constant psi . The structure of the wave is determined by a Floquet system of ordinary differential equations around the corresponding psi contour in the x,y plane, and the Floquet exponent gives the growth rate. This family of instabilities includes the Rayleigh centrifugal instability, the LeibovichStewartson columnar vortex instability, and the secondary instability of finiteamplitude waves in plane shear flows.
 Bayly, B. J., & Childress, S. (1987). Fast dynamo action in unsteady flows and maps in three dimensions. Physical Review Letters, 59(14), 15731576.More infoAbstract: Unsteady fast dynamo action is obtained in a family of stretchfoldshear maps applied to a spatially periodic magnetic field in three dimensions. Exponential growth of a mean field in the limit of vanishing diffusivity is demonstrated by a numerical method which alternates instantaneous deformations with molecular diffusion over a finite time interval. Analysis indicates that the dynamo is a coherent feature of the large scales, essentially independent of the cascade of structure to small scales. © 1987 The American Physical Society.
 Bayly, B. J., Goldhirsch, I., & Orszag, S. A. (1987). Independent degrees of freedom of dynamical systems. Journal of Scientific Computing, 2(2), 111121.More infoAbstract: Numerical evidence is presented demonstrating the possibility of obtaining reduced equations of motion for dynamical systems in terms of a small number of variables. Some implications of these results are discussed. © 1987 Plenum Publishing Corporation.
 Bayly, B. J. (1986). Fast magnetic dynamos in chaotic flows. Physical Review Letters, 57(22), 28002803.More infoAbstract: A generic structure is proposed for rapidly growing magnetic fields in a class of steady, incompressible flows which are everywhere chaotic with positive Liapunov exponent. In such a flow at high magnetic Reynolds number Rm, the magnetic field is approximately aligned everywhere with the dilating direction of the flow, and may have extremely complicated spatial structure. In the limit of infinite Rm, the leadingorder growth rate is bounded below by the positive Liapunov exponent of the flow; thus the field is a fast dynamo. © 1986 The American Physical Society.
 Bayly, B. J. (1986). Threedimensional instability of elliptical flow. Physical Review Letters, 57(17), 21602163.More infoAbstract: A theory is presented for Pierrehumbert's threedimensional shortwave inviscid instability of the simple twodimensional elliptical flow with velocity field u(x,y,z)=(Ey,E1x,0). The fundamental modes, which are also exact solutions of the nonlinear equations, are plane waves whose wave vector rotates elliptically around the z axis with period 2. The growth rates are the exponents of a matrix Floquet problem, and agree with those calculated by Pierrehumbert. © 1986 The American Physical Society.
 Bayly, B. J., & Yakhot, V. (1986). Positive and negativeeffectiveviscosity phenomena in isotropic and anisotropic Beltrami flows. Physical Review A, 34(1), 381391.More infoAbstract: A fieldtheoretic approach, analogous to Kraichnans directinteraction approximation, to the stability theory of complex threedimensional flows is developed. The longwavelength stability of a class of Beltrami flows in an unbounded, viscous fluid is considered. We examine two flows in detail, to illustrate the effects of strong isotropy versus strong anisotropy in the basic flow. The effect of the smallscale flow on the longwavelength perturbations may be interpreted as an effective viscosity. Using diagrammatic techniques, we construct the firstorder smoothing and directinteraction approximations for the perturbation dynamics. It is argued that the effective viscosity for the isotropic flow is always positive, and approaches a value independent of the molecular viscosity in the highReynoldsnumber limit; this flow is thus stable to longwavelength disturbances. The anisotropic flow has negative effective viscosity for some orientations of the disturbance, and is therefore unstable, when its Reynolds number exceeds 2. © 1986 The American Physical Society.
 Bayly, B. J. (1985). Heat transfer from a cylinder in a timedependent cross flow at low Peclet number. Physics of Fluids, 28(12), 34513456.More infoAbstract: The heat transfer from a constanttemperature cylinder in a uniform, timedependent cross flow at low Peclet number is considered. The time dependence is allowed to be strong, so that the velocity fluctuations may be comparable to, or larger than, the mean flow. The first nontrivial term in the Oseen approximation is calculated using matchedexpansion theory, and its physical significance is discussed. As an illustration, the timedependent heat transfer is calculated for a steady cross flow with large sinusoidal perturbations. © 1985 American Institute of Physics.
Presentations
 Pepper, E., Greivenkamp, J., Bayly, B. J., & Cuillier, C. A. (2017, June). Break Out of Your Library: Set Physics Loose with eBooks, Outreach, and Open Textbooks!. Special Libraries Association Annual Conference. Mesa, AZ: Special Libraries Association  Physics, Astronomy and Math Division.More infoThe 2017 Physics Roundtable will offer innovative ways to reach a wider library audience in the sciences. Discover how to utilize ebooks to make learning more portable, outreach to make science accessible and unforgettable, and open textbooks to lower the cost of higher education.