Turbulent  convection  flows  in  nature  are  often  found  organized  in  a  hierarchy  of  patterns  and structures. A prominent example is convection at the surface of the Sun. In my talk I will discuss our recent numerical simulations which address this pattern formation and the related transport within  simplified  simulation  models.  In  detail,  we  discuss  the  role  of  boundary  conditions  and compressibility. Convection processes at mesoscales typically have to be parametrized in global circulation  models.  In  the  talk,  we  will  also  report  our  recent  efforts  to  model  the  low-order statistics of buoyancy fluxes in reservoir computing and generative machine learning models. This includes efforts to implement these algorithms on noisy quantum computers.