The main coordinate system used by computational mesoscale models is a cartesian system XYZ with its origin at a point on a spherical earth model with the Z axis normal and exterior to the earth. The momentum equation solved by standard mesoscale models such as RAMS, MM5, ARPS or HOTMAC, considers that the gravity acceleration is opposite to the +Z direction. Some authors have pointed out that such an equation is valid on a small neighborhood of the origin. In the present work it is shown that the momentum equation is valid on a horizontal domain D(L)=2Lx2L with L smaller than 100 km. However, applications of mesoscale models have used a domain D(L) with L between 300 and 650 km (i) to reduce the error from lateral boundaries and (ii) to include the effects of some synoptic disturbances. It is shown that the use of the Newton's law of gravitation provides a momentum equation which is valid on any domain D(L). This is confirmed by an example which shows that the resulting momentum equations can yield the correct pressure field on the whole terrestrial sphere. Practical problems limit the use of the XYZ system to an L lower than 500km. In this case it is shown that a linear approximation of the Newton's law can be applied.

Some mesoscale models incorporate map projections into model equations to consider the earth curvature. This has motivated the use of such models on a domain D(L) with L=882, 1665 km. Formally, the governing equations from map projections are written in terms of a curvilinear coordinate system Xp, Yp, Zp, but it is shown that if such coordinates are approximated by X, Y, Z, respectively, the resulting momentum equation is valid on a region with L smaller than 100 km.