The earliest numerical models of cumulus convection were 2D (in a vertical plane) and simulated the lifecycle of a single cloud. With increases in computer power, simulations of cumulus ensembles became possible. Such simulations resolved individual cumulus clouds in a horizontal domain large enough to contain an ensemble of clouds. 2D CSRMs continue to be widely used for large-domain multiday simulations of cloud systems, and, most recently, have been used as a "super parameterization" in which a CSRM is embedded in each column of a GCM.
2D CSRMs are a reduced-dimensionality representation of a 3D flow. How do we get a 2D flow to do this? One way is to consider the 2D flow to represent the ensemble mean of a 3D flow subject to the constraint that the resolved cumulus convection is 2D. The utility of this approximation depends on the degree to which the constrained 3D flow acts like the actual fully 3D flow.
The 3D turbulence is parameterized in the 2D CSRM using ensemble mean closure. The goal is to represent the cascade of kinetic energy that occurs in a 3D turbulent flow. The eddy viscosity and kinetic enegy dissipation rate are set by the large (2D) eddy length and velocity scales, not those of the grid scale.
A 2D CSRM thus parameterizes the small, 3D eddies, while explicitly representing the large eddies. Such a two-scale representation is widely used in 1D boundary layer models, in which it is assumed that the small eddies are diffusive, while the large eddies perform non-local, countergradient transport.