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Galchen, T, Somerville RCJ.  1975.  On the Use of a Coordinate Transformation for Solution of Navier-Stokes Equations. Journal of Computational Physics. 17:209-228.   10.1016/0021-9991(75)90037-6   AbstractWebsite

The equations of fluid motion have been formulated in a generalized noncartesian, non-orthogonal coordinate system. A particular coordinate transformation, which transforms a domain with an irregular lower boundary into a cube, has been constructed. The transformed system, unlike the original one, has flat boundaries and homogeneous boundary conditions. Where the topography is flat, the original and transformed systems are identical, and extra terms do not appear. A finite difference scheme for solving the transformed equations has been constructed and will be described in a subsequent issue of this journal.

Pritchard, MS, Moncrieff MW, Somerville RCJ.  2011.  Orogenic Propagating Precipitation Systems over the United States in a Global Climate Model with Embedded Explicit Convection. Journal of the Atmospheric Sciences. 68:1821-1840.   10.1175/2011jas3699.1   AbstractWebsite

In the lee of major mountain chains worldwide, diurnal physics of organized propagating convection project onto seasonal and climate time scales of the hydrologic cycle, but this phenomenon is not represented in conventional global climate models (GCMs). Analysis of an experimental version of the superparameterized (SP) Community Atmosphere Model (CAM) demonstrates that propagating orogenic nocturnal convection in the central U.S. warm season is, however, representable in GCMs that use the embedded explicit convection model approach [i.e., multiscale modeling frameworks (MMFs)]. SP-CAM admits propagating organized convective systems in the lee of the Rockies during synoptic conditions similar to those that generate mesoscale convective systems in nature. The simulated convective systems exhibit spatial scales, phase speeds, and propagation speeds comparable to radar observations, and the genesis mechanism in the model agrees qualitatively with established conceptual models. Convective heating and condensate structures are examined on both resolved scales in SP-CAM, and coherently propagating cloud "metastructures" are shown to transcend individual cloud-resolving model arrays. In reconciling how this new mode of diurnal convective variability is admitted in SP-CAM despite the severe idealizations in the cloud-resolving model configuration, an updated discussion is presented of what physics may transcend the re-engineered scale interface in MMFs. The authors suggest that the improved diurnal propagation physics in SP-CAM are mediated by large-scale first-baroclinic gravity wave interactions with a prognostic organization life cycle, emphasizing the physical importance of preserving "memory" at the inner resolved scale.