Pattern Formation and Turbulence in Convection, the Legacy of Henri B\'enard.*

Just over a century ago a 26-year young physicist by the name of Henri B\'enard handed in his Ph.D. thesis, entitled {\it Les tourbillons cellulaires dans une nappe liquide}, at the Ecole Normale Sup\'erieure in Paris. In a fluid layer with a free upper surface and heated from below he observed and studied remarkably regular hexagonal patterns. Here I shall attempt to trace the developments in nonlinear physics, and especially in fluid mechanics, that have evolved from B\'enard's seminal experiments. As a result of the work of many, including Lord Rayleigh, Harold Jeffries (Sir Harold), W.V.R. Malkus and G. Veronis, and especially Fritz Busse (2000 Fluid-Dynamics-Prize recipient) and his long-term collaborator Richard Clever, a remarkably detailed understanding of the nature of convection in a shallow fluid layer between two solid horizontal confining surfaces and heated from below had been gained by the early 1970's. The bifurcation to convection is stationary and occurs at a temperature difference (in dimensionless form represented by the Rayleigh number $R$) and a wave number $k$ that are non-zero (Rayleigh, Jeffries). The bifurcation is supercritical to a pattern of rolls (Malkus and Veronis; Schl\"uter, Lortz, and Busse). Above onset there is a finite range in the $R-k$ plane, delimited by several interesting instabilities, over which the rolls are stable (Clever and Busse). This region, known now affectionately as the ``Busse Balloon", has been used during the last three decades to study both theoretically and experimentally numerous non-linear phenomena, including the role of thermal fluctuations near the bifurcation, the dynamics of pattern coarsening, various wave-number selection processes, spatio-temporal chaos, and spatially localized structures or ``pulses". In somewhat more recent times the range of $R$ has been extended up to $10^{14}$ times the critical value $R_c = {\cal O}(10^3)$ at onset and a richness of phenomena involving turbulent flows has been revealed and studied quantitatively. One of the particularly interesting issues amenable to study in this system has been the interaction between large-scale flow structures and the small-scale turbulent fluctuations; but there are many other aspects that have provided seemingly endless fascination for the researchers.