Geoscience Reference
In-Depth Information
• Eqs. 12.5-12.7 are Newton's second law (
F
ma
) in three dimensions as
applied to a fluid (i.e., including the advection terms), as derived in
chapter 6
.
• Eq. 12.8 is the irst law of thermodynamics (Eq. 5.3).
• Eq. 12.9 is the
continuity equation
, an expression of conservation of mass.
• Eq. 12.10 is the ideal gas law (Eq. 5.1).
• Eq. 12.11 is an equation for the conservation of water mass for a parcel of
air (see
chapter 9)
.
The model equations do not include explicit calculations for friction (
F
l
,
F
f
, and
F
z
), the diabatic heating rate (
Q
), precipitation (
P
), and evaporation
(
E
); these terms are represented by physical parameterizations. Parameter-
izations for friction are based on basic principles of momentum diffusion,
which occur on space scales much smaller than those resolved by GCMs
(discussed below). Convection may also occur on spaces scales smaller than
resolved and when this is the case—as it generally is—parameterizations are
used to calculate convective vertical velocities, water condensation rates, and
the resulting formation of clouds. The parameterization of convection and
condensational heating also yields the precipitation rate. Evaporation rates
are provided by coupling the AGCM with a land surface model, as discussed
below.
The AGCM's governing equations (Eqs. 12.5-12.11) are solved simultane-
ously at grid points for the
tendencies
(Eulerian time rates of change), which
are added to the most recent solution to advance the climate state in time. Typi-
cal horizontal grid spacings are 100-200 km with 30-50 vertical levels. Typical
time steps are about 10 minutes.
AGCMs use fixed sea surface temperatures from observations to specify the
lower boundary condition over the ocean, or a relatively simple heat balance
equation (e.g., Eq. 5.22) to represent a mixed-layer, or slab, ocean with heat
capacity but no circulation. A more complete treatment of the climate system
requires coupling the AGCM with its ocean counterpart.
OCEAN GCMS
Ocean GCMs
(OGCMs) are governed by a set of equations that are very simi-
lar to those used in AGCMs (Box. 12.1) but with some important differences.
Because the ocean cannot be treated as an ideal gas (
chapter 5
), Eq. 12.10 is
replaced by an equation of state for the ocean that includes the dependence
of water density on salinity in addition to temperature and pressure. Also, Eq.
12.11, for the conservation of water vapor, is replaced by an equation for the
conservation of salt,
2
()
ρ
S
v
=− −+−
u
$
d
() () () ( ,
ρ
S
fP fE fR
(12.12)
2
t
where
S
is salinity and the functions
f
(
P
),
f
(
E
), and
f
(
R
) represent the influence
of precipitation, evaporation, and runoff (as from rivers, for example), respec-
tively, on the salinity distribution.
