1.2 WaveEquations in Geophysical Fluid Dynamics
23
be linearized and modificd to obtain an energy-conservative system of anelastic
equations. As a first step toward developing such a system, the thermodynamic
variables are decomposed into a vertically varying reference state and a perturbation. This decomposition is also quite useful outside the context of the anelastic
equations because in many geophysical fluids the gravitational acceleration and
the vertical pressure gradient are nearly in balance. Both numerical accuracy and
physical insight can be then be enhanced by splitting the pressure and density
fields into steady hydrostatically balanced vertical profiles and finite-amplitude
perturbations about those reference profiles such that
p(x , y, Z, t) = p(z) + p'(x , y, z, t),
p(x , y, z, t) = p(z) + p'(x, y, z. t),
dp
_
-=-pg.
dz
After removing the hydrostatically balanced component of the pressure, the momentum equation (1.31) may be written without approximation as
dv
-
1 ,
p'
+ -'Vp = -g-k.
dt
p
p
(1.57)
If the pressure gradients in the momentum equation are expressed in terms of rr
and (), the hydrostatic reference state is removed by defining
rr(x , y, z, r) = n(z) + rr'(x, y , z, r),
()(x, y, Z, t) = e(z) + O'(», y, z. t),
-dn
cp()d; = -g ,
in which case (1.37) becomes
(1.58)
dv
-d + cp()'Vrr' = g=k.
e'
(1.59)
t
()
dv
-
1 ,
p'
+ -'Vp = -g-k,
dt
Po
Po
(1.60)
The term on the right side of either (1.57) or (1.59) represents a buoyancy force.
Note that since no approximations have been introduced in these equations, the
pressure gradient terms in (1.57) and (1.59) remain nonlinear.
In addition to the previously discussed modifications to the mass continuity
equation, the Boussinesq and anelastic approximations include additional simplifications to the momentum equations that linearize the pressure gradient terms in
(1.57) and (1.59). The form of the Boussinesq approximation that is most common in geophysical fluid dynamics neglects the effects ofdensity variations on the
mass balance in the continuity equation and on inertia in the momentum equations, but includes the effect ofdensity variations on buoyancy forces (Gill 1982,
p. 130). Letting Po be a constant reference density, the Boussinesq form of the
momentum equations may be written
23
be linearized and modificd to obtain an energy-conservative system of anelastic
equations. As a first step toward developing such a system, the thermodynamic
variables are decomposed into a vertically varying reference state and a perturbation. This decomposition is also quite useful outside the context of the anelastic
equations because in many geophysical fluids the gravitational acceleration and
the vertical pressure gradient are nearly in balance. Both numerical accuracy and
physical insight can be then be enhanced by splitting the pressure and density
fields into steady hydrostatically balanced vertical profiles and finite-amplitude
perturbations about those reference profiles such that
p(x , y, Z, t) = p(z) + p'(x , y, z, t),
p(x , y, z, t) = p(z) + p'(x, y, z. t),
dp
_
-=-pg.
dz
After removing the hydrostatically balanced component of the pressure, the momentum equation (1.31) may be written without approximation as
dv
-
1 ,
p'
+ -'Vp = -g-k.
dt
p
p
(1.57)
If the pressure gradients in the momentum equation are expressed in terms of rr
and (), the hydrostatic reference state is removed by defining
rr(x , y, z, r) = n(z) + rr'(x, y , z, r),
()(x, y, Z, t) = e(z) + O'(», y, z. t),
-dn
cp()d; = -g ,
in which case (1.37) becomes
(1.58)
dv
-d + cp()'Vrr' = g=k.
e'
(1.59)
t
()
dv
-
1 ,
p'
+ -'Vp = -g-k,
dt
Po
Po
(1.60)
The term on the right side of either (1.57) or (1.59) represents a buoyancy force.
Note that since no approximations have been introduced in these equations, the
pressure gradient terms in (1.57) and (1.59) remain nonlinear.
In addition to the previously discussed modifications to the mass continuity
equation, the Boussinesq and anelastic approximations include additional simplifications to the momentum equations that linearize the pressure gradient terms in
(1.57) and (1.59). The form of the Boussinesq approximation that is most common in geophysical fluid dynamics neglects the effects ofdensity variations on the
mass balance in the continuity equation and on inertia in the momentum equations, but includes the effect ofdensity variations on buoyancy forces (Gill 1982,
p. 130). Letting Po be a constant reference density, the Boussinesq form of the
momentum equations may be written