1.2 Wave Equations in Geophysical Fluid Dynamies
21
include both the preceding and additional approximations in the momentum equations that will be discussed in connection with (1.60). The latter definition, encompassing approximations to both the mass continuity and momentum equations,
appears to be consistent with the actual approximations employed by Boussinesq
(1903, pp. 157 and 174), and will be the form of the Boussinesq approximation
referred to throughout this book.
A second approximation to the full compressible continuity equation is anelastic compressibility (Ogura and Phillips 1962; Lipps and Hemler 1982)
v· (pv) = 0,
(1.52)
in whieh the density involved in the mass budget is a steady reference-state density p(z) that varies only along the coordinate axis parallel to the gravitational
restoring force. A third approximation is pseudo-incompressibility (Durran 1989)
-
ap
+
A
'il . (pv) = 0,
at
(1.53)
in whieh pis determined by the time-varying potential temperature and the pressure in a steady reference state ß(x , y, z) via the equation of state
_
P = PO
(R A )CP/cv
-pO
PO
The pseudo-incompressible approximation neglects the influence of perturbation
pressure on perturbation density in the mass budget. According to the preceding
definition of p, the term aNat in (1.53) is entirely determined by
The
pseudo-incompressible continuity equation may be written in the obviously diagnostie form
v · (pOv) = 0
(1.54)
by using the thermodynamie equation (1.33) to eiiminate ao/at from (1.53) and
defining steady reference fields of density p(x , y, z) and potential temperature
o(x, y, z) such that the reference fields satisfy the equation of state,
_ (R __t:
P = Po -pO
PO
Note that if Fe represents any thermal forcing or viscous terms that might appear
on the right side of the thermodynamie equation in more general appiications,
(1.53) is unchanged but (1.54) becomes
v . (pOv) = pFe.
The pseudo -incornpressible system can be rigorously derived through scale analysis by assuming that the Mach number (U /cs) and the perturbation of the total
pressure about the reference pressure, ß, are both small (Durran 1989).
21
include both the preceding and additional approximations in the momentum equations that will be discussed in connection with (1.60). The latter definition, encompassing approximations to both the mass continuity and momentum equations,
appears to be consistent with the actual approximations employed by Boussinesq
(1903, pp. 157 and 174), and will be the form of the Boussinesq approximation
referred to throughout this book.
A second approximation to the full compressible continuity equation is anelastic compressibility (Ogura and Phillips 1962; Lipps and Hemler 1982)
v· (pv) = 0,
(1.52)
in whieh the density involved in the mass budget is a steady reference-state density p(z) that varies only along the coordinate axis parallel to the gravitational
restoring force. A third approximation is pseudo-incompressibility (Durran 1989)
-
ap
+
A
'il . (pv) = 0,
at
(1.53)
in whieh pis determined by the time-varying potential temperature and the pressure in a steady reference state ß(x , y, z) via the equation of state
_
P = PO
(R A )CP/cv
-pO
PO
The pseudo-incompressible approximation neglects the influence of perturbation
pressure on perturbation density in the mass budget. According to the preceding
definition of p, the term aNat in (1.53) is entirely determined by
The
pseudo-incompressible continuity equation may be written in the obviously diagnostie form
v · (pOv) = 0
(1.54)
by using the thermodynamie equation (1.33) to eiiminate ao/at from (1.53) and
defining steady reference fields of density p(x , y, z) and potential temperature
o(x, y, z) such that the reference fields satisfy the equation of state,
_ (R __t:
P = Po -pO
PO
Note that if Fe represents any thermal forcing or viscous terms that might appear
on the right side of the thermodynamie equation in more general appiications,
(1.53) is unchanged but (1.54) becomes
v . (pOv) = pFe.
The pseudo -incornpressible system can be rigorously derived through scale analysis by assuming that the Mach number (U /cs) and the perturbation of the total
pressure about the reference pressure, ß, are both small (Durran 1989).