Turbulence in vertically averaged convection

The vertically averaged velocity of the 3D Rayleigh-Bénard problem is analyzed and numerically simulated. This vertically averaged velocity satisfies a 2D incompressible Navier-Stokes system with a body force involving the 3D velocity. A time average of this force is estimated through time averages of the 3D velocity. Relations similar to those from 2D turbulence are then derived. Direct numerical simulation of the 3D Rayleigh-Bé nard is carried out to test how prominent the features of 2D turbulence are for this Navier-Stokes system.