7.5 The Hydrostatic Approximation
373
the hydrostatic relation
ap
-
az
= -pg.
(7.91)
The hydrostatic system is not a hyperbolie system of partial differential equations''
beeause there is no prognostie equation for the vertical velocity. The numerieal
solution of the hydrostatie system requires the evaluation of time-independent
equations, sueh as (7.91), at every time step. These diagnostic equations ean be
evaluated with mueh less computational effort than that required to solve the diagnostic Poisson equation for the pressure in the Boussinesq system.
The hydrostatie approximation eliminates sound waves, although as will be
discussed below, the hydrostatic approximation does not remove all horizontally
propagating acoustic modes . In those large-seale geophysical applieations where
the numerieal resolution along the vertical coordinate is much finer than the horizontal resolution, explieit finite-difference approximations to the hydrostatic system can be integrated much more efficiently than eomparable approximations
to either the nonhydrostatic Boussinesq equations or the nonhydrostatic compressible equations. The eonsiderable improvement in model efficieney associated
with the use of the hydrostatic goveming equations does not, however, apply to
semi-implieit models beeause these models ean easily be modified to compute
semi-implicit approximations to the full nonhydrostatic eompressible equations
without signifieantly increasing the computational overhead (Culien 1990; Tanguay et al. 1990).
The influence of the hydrostatic approximation on wave propagation and the
stability eriteria for explicit finite-difference approximations to the hydrostatic
equations may be detennined by examining solutions to the linearized hydrostatie
system. Small-amplitude hydrostatically balaneed perturbations in the x-z plane
about a resting isothennally stratified basic state are govemed by the system
au ap
- + - = 0
at ax '
(:z+r) P =b ,
ab
-+N
at
2w=0 ,
(7.92)
(7.93)
(7.94)
- er
at + c 2 [au - + ( - - a - r )] W = 0,
s ax az
(7.95)
where u, w, b, P, I' , and cs are defined by (7.35)-(7.37). Waves ofthe form
(u , w, b, P) = (uo, Wo, bo, Po)ei(kxHz-wnßl)
8There has been some concem about the well-posedness of initial-boundary value problems involving the hydrostatic equations (Oliger and Sundstr öm 1978). It is not clear how to reconcile these
concems with the successful forecasts obtained twice daily at several operational centers for at least
two decades using Iimited-area weather prediction models based on the hydrostatic goveming equations.
373
the hydrostatic relation
ap
-
az
= -pg.
(7.91)
The hydrostatic system is not a hyperbolie system of partial differential equations''
beeause there is no prognostie equation for the vertical velocity. The numerieal
solution of the hydrostatie system requires the evaluation of time-independent
equations, sueh as (7.91), at every time step. These diagnostic equations ean be
evaluated with mueh less computational effort than that required to solve the diagnostic Poisson equation for the pressure in the Boussinesq system.
The hydrostatie approximation eliminates sound waves, although as will be
discussed below, the hydrostatic approximation does not remove all horizontally
propagating acoustic modes . In those large-seale geophysical applieations where
the numerieal resolution along the vertical coordinate is much finer than the horizontal resolution, explieit finite-difference approximations to the hydrostatic system can be integrated much more efficiently than eomparable approximations
to either the nonhydrostatic Boussinesq equations or the nonhydrostatic compressible equations. The eonsiderable improvement in model efficieney associated
with the use of the hydrostatic goveming equations does not, however, apply to
semi-implieit models beeause these models ean easily be modified to compute
semi-implicit approximations to the full nonhydrostatic eompressible equations
without signifieantly increasing the computational overhead (Culien 1990; Tanguay et al. 1990).
The influence of the hydrostatic approximation on wave propagation and the
stability eriteria for explicit finite-difference approximations to the hydrostatic
equations may be detennined by examining solutions to the linearized hydrostatie
system. Small-amplitude hydrostatically balaneed perturbations in the x-z plane
about a resting isothennally stratified basic state are govemed by the system
au ap
- + - = 0
at ax '
(:z+r) P =b ,
ab
-+N
at
2w=0 ,
(7.92)
(7.93)
(7.94)
- er
at + c 2 [au - + ( - - a - r )] W = 0,
s ax az
(7.95)
where u, w, b, P, I' , and cs are defined by (7.35)-(7.37). Waves ofthe form
(u , w, b, P) = (uo, Wo, bo, Po)ei(kxHz-wnßl)
8There has been some concem about the well-posedness of initial-boundary value problems involving the hydrostatic equations (Oliger and Sundstr öm 1978). It is not clear how to reconcile these
concems with the successful forecasts obtained twice daily at several operational centers for at least
two decades using Iimited-area weather prediction models based on the hydrostatic goveming equations.