PHYSICAL REVIEW E 92, 013005 (2015) Bicritical states in temperature-modulated Rayleigh-B´ enard convection Jitender Singh, 1 , * Renu Bajaj, 2 , † and Puneet Kaur 1 , ‡ 1 Department of Mathematics, Guru Nanak Dev University, Amritsar 143005, Punjab, India 2 Department of Mathematics, Panjab University, Chandigarh 160014, India (Received 13 February 2015; published 6 July 2015) We investigate the Rayleigh-B´ enard convection under sinusoidally varying temperatures of the horizontal rigid planes bounding a laterally infinite fluid layer for the bicritical states. The problem is analogous to the well studied Faraday instability and Rayleigh-B´ enard convection under gravity modulation. Under modulation, the neutral instability curve is found to alternate between the conventional harmonic and subharmonic tongues in the space of the dimensionless wave number of disturbance and the control parameter. The transition between harmonic and subharmonic critical instability responses is found to occur via a bicritical state, where the two instability responses coexist with different wave numbers. These bicritical states are found to depend upon the modulation parameters and the Prandtl number. DOI: 10.1103/PhysRevE.92.013005 PACS number(s): 44.27.+g, 47.15.Fe, 47.20.Bp I. INTRODUCTION The Rayleigh-B´ enard convection (henceforth referred to as RBC) is the buoyancy-driven instability in an initially quiescent, laterally infinite, viscous fluid layer confined be- tween two horizontal planes subjected to a sufficiently large vertical temperature gradient. The instability is usually found to exhibit definite fluid pattern in the form of polygonal plane tiling. The RBC has become a paradigm of pattern-forming systems because of its rich dynamics and practical relevance, which makes it a common interest among the diverse areas, such as geophysics, atmospheric science, biomedical science, engineering, and mathematics [1,2]. When the initially quiescent fluid layer is subjected to a time-periodic forcing, the instability appears in the form of a definite time-periodic fluid pattern, which oscillates either with the forcing frequency (harmonic oscillation) or half the forcing frequency (subharmonic oscillation). The instability induced in this way is called the parametric instability. The harmonic and subharmonic oscillations arise in several contexts, such as the motion of the simple pendulum, electronic oscillators, convection in earth’s mantle, atmosphere, sun, and stars. However, the study of the instabilities in fluid mechanics dates back to the experiments of Faraday (1831) who observed an instability in a plane-free surface of the vertically vibrating fluid layer, which is commonly known as the Faraday instability. His observations were supported by the mathematical work of Lord Rayleigh [3]. For a detailed account of the Faraday instability, the reader may refer to the excellent review by Miles and Henderson [4]. A similar situation arises when the horizontal fluid layer is excited via sinusoidal oscillation in the temperatures of one or both of the bounding planes. The instability is known as the temperature modulated Rayleigh-B´ enard convection (hereafter referred to as TMRBC). The instability is known * https://sites.google.com/site/sonumaths2/; sonumaths@gmail. com † rbajaj@pu.ac.in ‡ puneet.kaur99@gmail.com to manifest itself in the form of a polygonal pattern when the temperature gradient across the fluid layer exceeds a certain critical value. At this stage, the entire fluid pattern is found to oscillate time periodically, either harmonically or subharmon- ically, depending upon the extent of modulation [1,5,6]. Regardless of the different origins, TMRBC, the Faraday instability, and the RBC under gravity modulation share identical oscillating fluid patterns in two dimensions, which have been observed in the fluid layer theoretically and verified experimentally. The patterns include rolls, triangles, squares, hexagons, and octagonal and dodecagonal plane tiling of the fluid layer. Some of the featured patterns observed experimentally in the Faraday instability correspond to “bi- critical states” in an appropriate parametric space, where the harmonic and subharmonic oscillations coexist with distinct wave numbers [7]. The bicritical situations commonly occur in the Faraday instability and RBC under gravity modulation, both experimentally as well as theoretically. Therefore, the literature is wide and for a quick review, the reader may refer to the work in Miles and Henderson [4], Edwards and Fauve [8], Volmer and M¨ uller [9], and references cited therein. Also, near the onset of a bicritical state in a pattern-forming system, a competition between the nonlinear harmonic and subharmonic modes results in new resonant “superlattice patterns” with rich dynamics, which makes it an important instability phenomenon from experimental and theoretical point of view [10,11]. Like the Faraday instability and RBC under gravity modulation, an extensive theoretical and experimental work on TMRBC has also been reported in the literature, where the theoretical inferences are generally found to be in good agreement with the laboratory experiments [6]. However, in all the previous research on TMRBC, apart from investigation on the nature of the instability, the emphasis has been on obtaining the critical onset of the instability via the linear approximation, the local-nonlinear analysis, or evaluation of the global nonlinear stability boundary [2,12–19]. To the best of the authors’ knowledge, the investigation on existence of bicritical states in TMRBC has largely remained missing in the earlier experimental and theoretical research on TMRBC, which forms the objective of our present work. 1539-3755/2015/92(1)/013005(7) 013005-1 ©2015 American Physical Society