A framework for suppressing triad interaction in nonlinear dynamical systems: application to multiple states in two-dimensional Rayleigh–Bénard convection

Nonlinear dynamical systems often allow for multiple statistically stationary states for the same values of the control parameters. Herein, we introduce a framework that selectively eliminates specific nonlinear triad interactions, thereby suppressing emergence of a particular state, and enabling the emergence of another. The methodology is applied to yield the multiple convection-roll states in two-dimensional planar Rayleigh–Bénard convection (e.g. Wang et al., 2020, Phys. Rev. Lett. , vol. 125, 074501) in the turbulent regime. The intrusive framework presented here is based on the observation that the characteristic wavenumber associated with the mean horizontal size of the convection rolls mediates triadic scale interactions resulting in both kinetic energy and temperature variance cascades that are dominant energy transfer processes in a statistically stationary state. Suppression of these cascades mediated by a candidate wavenumber hinders the formation of the convection rolls at that wavenumber. Consequently, convection rolls are formed at another candidate wavenumber which is allowed to mediate energy to establish the cascade processes. In case no stable convection-roll states are possible, this technique does not tend to yield any convection rolls, making it a suitable method for discovering multiple states. Whereas in previous investigations the signature of different states in the initial condition in simulations yielded the multiple states, the present method alleviates such dependence of the emergence of multiple states on initial conditions. It is also demonstrated that accurate predictions of statistical quantities, such as Nusselt number and volume-averaged Reynolds numbers, can also be obtained using this method. The convection-roll states yielded using this technique may be used as initial conditions for direct simulations quickly converging to the target roll state without taking long convergence routes involving state transitions. Additionally, because only one state can possibly emerge in each simulation, this technique can empirically designate each of the multiple states with respect to their stability.