A Jacobian-free Newton-Krylov solver for determination of scaling laws in coherent Rayleigh-B\'{e}nard convection

Computational studies of \emph{coherent} Rayleigh-B\'{e}nard convection in a two-dimensional channel with no-slip top and bottom walls are performed in order to determine scaling laws for a range of Rayleigh ($Ra$) and Prandtl ($Pr$) numbers. Since these coherent states are unstable, a Jacobian-free Newton-GMRES algorithm is developed. This approach allows us to determine scaling of the Nusselt number ($Nu$) with $Ra$ by tracking unstable solutions to the Boussinesq equations. Scaling laws are presented for the primary solution that bifurcates from the conducting state at $Ra \sim 1708$, becomes unstable in a Hopf bifurcation at $Ra \sim 5.4\times 10^{4}$ but have been computed up to $Ra \sim 5\times 10^{6}$. We also determine scaling laws for the optimal heat transport up to $Ra\sim 10^{8}$. Mechanisms for the observed behavior are discussed including the relationship between the optimal solution and the primary solution as well as the effect of $Pr$. We explore properties of the algorithm and review its potential as a tool in determining scaling laws for thermal convection as well as some areas for improvement. Extensions of this work to three-dimensional Rayleigh-B\'{e}nard convection will be discussed.