Closure of equations obtained via the HSP-VAT averaging techniques is an old and standing problem. There is the substantial literature exists with regard of treatment of additional terms or coefficients written for Upper Scale formulated problems. At the same time it is hard to find works (apart of ours) where the closure issues is directly connected to a problem. I mean where the closure is directly connected to physics and morphology of the stated problem. Generally, the point is how the averaged problem was formulated. The HSP-VAT formulation makes the closure really connected to a problem's morphology. Here the closure can be traced straight from the medium's morphology.
While other workers in the area of HSP-VAT research seems do not believe into the reality, into the true scale communications between the physical and mathematical models for different scale, I was a fierce advocate of direct connection between those not only in writing the VAT equations, but in closure and in the simulation using the exact Lower Scale equations just being used for developing the Upper Scale equations.
The very important issue in the separate HSP-VAT's scales mathematical models consideration and closure is that the direct interconnection, communication of physical characteristics and properties becomes unattached. And we can not say or provide the direct dependency or formulae for the separate scale properties! Meanwhile, this is often one of the main goals to state the complicated scaled model!
That is why, we used to go after the development of the Closure Problems in HSP-VAT as based on the idea of the both scales mathematical statements involved. This gives the true mathematical and physical interconnection, often the exact interconnection of problem's properties on both scales.
We describe below and give the reference publications where the features for general methodologies for the HSP-VAT one (the Upper) and the Two scale Closure and few examples of problems have been closed and solved exactly on both scales.