A computationally efficient multi-scale planar-averaging framework for urban areas is developed, which enables efficient computation of coarse-grained velocity and scalar fields. We apply the multi-scale framework to a large-eddy simulation of an idealised heterogeneous urban environment of 512 buildings based on a typical London height distribution. We observe that for this geometry, the characteristic urban lengthscale \(\ell \approx 50\) m, which is the averaging lengthscale L at which as much variance in the mean flow is resolved as is unresolved. For \(L>400\) m, the statistics become approximately homogeneous, suggesting that non-building-resolving numerical weather prediction (NWP) models can be applied without modification at resolutions of 400 m and above for the case under consideration. We derive the multi-scale planar- and Reynolds-averaged momentum equation and show that for neutral cases, NWP models require parameterisation of the distributed drag and the unresolved turbulence and dispersive stress. An a priori analysis reveals that the drag parameterisation from Sützl et al. (Bound-Layer Meteorol 178:225–248, 2020) holds reasonably well for resolutions L above 200 m. Below this value, the problem becomes inhomogeneous and the parameterisation works less well. The unresolved stresses are well represented by a \(k-\omega \) closure with a value of \(\omega =0.4\, \text {s}^{-1}\) . However, an even more accurate closure can be derived from the Sützl drag parameterisation that does not require further turbulence information.
