Abstract

 We discuss the initial value problem for Rayleigh-B´enard convective adjustment (RBCA) in a vertical plane, using the Euler-Boussinesq equations in the vorticity representation, neglecting viscosityand thermal diffusivity. This is a non-dissipative, initial-value version of the equations studied in². The Hamiltonian structure of the model is used to characterise the equilibrium solutions of thenon-dissipative system and derive their corresponding Taylor-Goldstein equations for linear instability. Stochastic advection by Lie transport (SALT) is introduced by following the approach of [1, 3].The SALT equations enable uncertainty quantification and admit data assimilation methods based onparticle filtering that can reduce the uncertainty in coarse-grained computational simulations of convective adjustment. The Lagrangian Averaged (LA) SALT equations for the RBCA initial value problemare discussed. The expectation equations of LA SALT for RBCA are similar in appearance to theoriginal dissipative Oberbeck-Boussinesq equations for diffusive, viscous dynamics of Rayleigh-B´enardconvection. Finally, deterministic dynamical equations for the covariances and higher moments of thefluctuations away from the expected solution are derived