 
Summary: MATHEMATICAL JUSTIFICATION OF THE HYDROSTATIC
APPROXIMATION IN THE PRIMITIVE EQUATIONS OF
GEOPHYSICAL FLUID DYNAMICS #
PASCAL AZ ’
ERAD + AND FRANCISCO GUILL ’
EN #
SIAM J. MATH. ANAL. c
# 2001 Society for Industrial and Applied Mathematics
Vol. 33, No. 4, pp. 847859
Abstract. Geophysical fluids all exhibit a common feature: their aspect ratio (depth to hori
zontal width) is very small. This leads to an asymptotic model widely used in meteorology, oceanog
raphy, and limnology, namely the hydrostatic approximation of the timedependent incompressible
NavierStokes equations. It relies on the hypothesis that pressure increases linearly in the vertical
direction. In the following, we prove a convergence and existence theorem for this model by means
of anisotropic estimates and a new timecompactness criterium.
Key words. NavierStokes equations, shallow domains, geophysical fluid dynamics, hydrostatic
approximation, singular perturbation, compactness criterium, asymptotic analysis
AMS subject classifications. 35Q30, 35B40, 76D05, 34C35
PII. S0036141000375962
1. Introduction. Atmospheric flow in meteorology, water flow in oceanography,
