Meteorol. Atmos. Phys. 63, 3-14 (1997) Meteorology, and Atmospheric Physics 9 Springer-Verlag 1997 Printed in Austria U.S. National Centers for Environmental Prediction/UCAR Visitor Research Program, Camp Spring, MD, U.S.A. Dynamics of Limited-Area Models: Formulation and Numerical Methods* F. Mesinger Received 28 March 1996 Revised August 18, 1996 Summary Following a few historical remarks, approximations used in formulating the "dynamics" of limited-area and variable resolution atmospheric forecasting models are reviewed. Particular attention is given to current efforts to relax or remove the hydrostatic approximation. Turning to numerical methods used in discretizing the equations, an attempt is made to record recep* v3rk and to clarify the motivation for the various approaches being followed by different modeling centers. Topics commented upon include: semi-Lagrangian methods, numerical for- mulation of nonhydrostatic models, resolution, the eta (step-mountain) vs sigma or isentropic/sigma vertical coordinate, choice of the vertical grid, numerics of the propagation of gravity waves, and the box-average vs point- sample treatment of predicted variables. It is finally pointed out that the extraordinary diversity of roads being taken shows that a lot remains to be discovered as to what possible rewards may be found in exploring one or the other of the principles underlying the methods being developed. 1. Background: Limited-area Modeling For a review a few historical remarks seem in order. But historical remarks on what? While *This is a revised and updated text of an invited lecture given within the "One-Day Intensive Course" of the Inter- national Workshop on Limited-Area and Variable Resolution Models, Beijing, China, 23-27 October 1995. there is little to be debated as to what were the milestones marking the beginnings of numerical weather prediction, when it comes to the beginnings of atmospheric "limited area" or "regional" modeling, a question of what is actually meant comes up. The first numerical forecast of Richardson (1922) and the work on all subsequent models at the outset of NWP was of course limited area modeling in the literal sense. The term "limited area" came however into widespread use at a later stage, following the establishment of hemispheric and global models. Clearly, some ambiguity remains. Do we have in mind as the beginnings of limited area modeling the first successful efforts aimed at the simulation of meso-alpha scale weather phenomena, such as the "frontal rain" emphasized in Bushby and Timpson's (1967) paper? Or should a distin- guishing feature be that of running a numerical model with forecast lateral boundary conditions, in research and then in an operational mode? Taking the last possibility, of an operational use of forecast boundary conditions, it has now been about a quarter of a century since the beginnings of atmospheric regional modeling. One milestone of this nature has recently been celebrated at the U.S. National Meteorological Center (NMC; now National Centers for Environmental Prediction, NCEP) by the "LFM birthday party" (Anon- ymous, 1993), honoring the coming of age in