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Obituary: Charles Doering

By John D. Gibbon and Leonard Sander

Charles Rogers Doering, 1956-2021. Photo courtesy of the University of Michigan.
Charles Rogers Doering, an eminent applied mathematician and physicist, died on May 15, 2021 after a year-long, courageous fight with cancer. He was the Nicholas D. Kazarinoff Collegiate Professor of Complex Systems, Mathematics, and Physics, as well as director of the Center for the Study of Complex Systems at the University of Michigan. Charlie was a SIAM Fellow and a Fellow of the American Physical Society. He was particularly active in the SIAM Activity Group on Analysis of Partial Differential Equations and on SIAM’s Diversity Advisory Committee. Throughout his career, he held Guggenheim and Simons Fellowships and received a Presidential Young Investigator Award, a Fulbright Scholarship, and a Humboldt Research Prize. His untimely passing at the age of 65 has caused deep sorrow among his many friends and colleagues.

Charlie was born in Philadelphia, Pa., and raised in Schenectady, NY. He received a B.S. from Antioch College (Ohio), an M.S. from the University of Cincinnati, and a Ph.D. in mathematical physics at the University of Texas at Austin under the direction of Cecille de Witt. Between his stints at Cincinnati and Texas, Charlie spent time on the road with his old college band attempting to “make the scene” as a rock musician. However, as he once wryly admitted, the need to eat became increasingly urgent and he returned to the more conventional life of a physicist. Upon graduating from Texas, Charlie became a postdoc at the Center for Nonlinear Studies (CNLS) at Los Alamos National Laboratory before joining Clarkson University’s Physics Department. After nine years at Clarkson, during which he rose to the rank of full professor, he returned to Los Alamos as deputy director of the CNLS. In 1996, Charlie joined the mathematics faculty at the University of Michigan.

Charlie’s work was wide ranging and fiercely interdisciplinary. His physical insight—combined with great technical skill—allowed him to reduce complex problems to more tractable models. In complex systems, his main work lay in the stochastic processes that occur in physics, chemistry, biology, and ecology. Charlie was also active and influential in the study of the stochastic dynamics of low-dimensional systems. His seminal paper on resonant activation is his most-cited work [6], and his clever paper on the stochastic ratchet and the significance of colored noise is fundamental to that field [8]. Charlie was a true polymath, and his papers on chemical kinetics, neuroscience, and mathematical ecology have all had considerable impact. He was a generous and inspiring collaborator, and his collaborations were numerous. Indeed, he successfully led the Center for the Study of Complex Systems at Michigan for six years.

Charlie was both a skilled and intuitive physicist and a fine applied analyst in the mathematical sense. During his two periods at Los Alamos, he became intensely interested in the dynamics of turbulent fluid flows. His resulting book on the subject with J.D. Gibbon is an indispensable source for these problems and has served as an introduction to the field for many young mathematicians and physicists [7]. 

Charlie’s most prominent work was with Peter Constantin in the form of a masterly series of papers [1-5]. In these papers, the two developed what is now called the “background field method” to handle the problem of flows with boundaries. For instance, in Rayleigh-Bénard (RB) convection, a key question asks how the Nusselt number (\(\textrm{Nu}\))—a measure of the heat flux between the two boundaries—scales with the Rayleigh number (\(\textrm{Ra}\)). In 1954, Willem Malkus suggested that the heat transport in an RB flow should be maximized. Lou Howard and Fritz Busse significantly developed this idea in subsequent years using variational methods. However, Howard found that his derivation of the Euler-Lagrange equations for the maximal heat flux relied on an additional assumption of stationary statistics. The work of Doering and Constantin put the variational method on a rigorous Navier-Stokes footing and removed the need for stationary statistics. They were also able to derive a rigorous upper bound of the form \(\textrm{Nu} \le \textrm{Ra}^{1/2}\); the ultimate state would occur if this bound were saturated. Moreover, they derived an improved bound of \(\textrm{Nu} \le \textrm{cRa}^{3/8}\) — provided that the optimal background profile has a nondegenerate ground state. The background field method has since become a standard tool in variational methods of fluid convection.

Charlie’s work in convection brought him into contact with the Walsh Cottage group at Woods Hole Oceanographic Institution, which is home to an influential summer graduate program in geophysical fluid dynamics. Over the last 20 years, Charlie became a regular attendee, committee man, leader, and enthusiastic member of the group’s softball team. A few years ago, an over-vigorous attempt to make second base led to the rupture of both of his patella tendons, the price of which was several months in a wheelchair. Charlie’s enthusiasm was boundless and his quotations from the legendary Yogi Berra were extensive, but on his skills as a softball coach we will remain diplomatically silent. 

Charlie was not only a remarkable teacher and lecturer — he was a master of blackboard technique as well. He loved to explain things; for Michigan’s Saturday Morning Physics TV program, he once attempted to convey to a lay audience the fascination of turbulent transport and RB convection (he almost succeeded). Charlie had remarkable energy and an irresistible, impish sense of humor. Moreover, as a football superfan he had no equal — this prestige led to considerable fame outside the scholarly community. On football Saturdays, he traveled around Ann Arbor wearing an astonishing maize suit, a blue tie (the Michigan colors are maize and blue), a white fedora, white gloves, and saddle oxfords. An unforgettable photo is available online. The irony of a famous mathematician dressed this way even made The New York Times

Well, we think he meant it ironically, but with Charlie one never knew. 


References 
[1] Constantin, P., & Doering, C.R. (1995). Variational bounds in dissipative systems. Physica D, 82, 221.
[2] Constantin, P., & Doering, C.R. (1996). Heat transfer in convective turbulence. Nonlinearity, 9, 1049.
[3] Doering, C.R., & Constantin, P. (1992). Energy dissipation in shear driven turbulence. Phys. Rev. Lett., 69, 1648.
[4] Doering, C.R., & Constantin, P. (1994). Variational bounds on energy dissipation in incompressible flows: Shear flow. Phys. Rev. E, 49, 4087.
[5] Doering, C.R., & Constantin, P. (1996). Variational bounds on energy dissipation in incompressible flows: III Convection. Phys. Rev. E, 53, 5957.
[6] Doering, C.R., & Gadoua, J.C. (1992). Resonant activation over a fluctuating barrier. Phys. Rev. Lett., 69, 2318.
[7] Doering, C.R., & Gibbon, J.D. (1995). Applied analysis of the Navier-Stokes equations. In Cambridge texts in applied mathematics. Cambridge, U.K.: Cambridge University Press.
[8] Doering, C.R., Horsthemke, W., & Riordan, J. (1994). Nonequilibrium fluctuation-induced transport. Phys. Rev. Lett., 72, 2984.

John D. Gibbon is a professor of applied mathematics at Imperial College London. Leonard M. Sander is an emeritus professor of physics and complex systems at the University of Michigan.

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