Geography Reference
In-Depth Information
Subtracting (2.38) from (2.37), we obtain
ρ
De
Dt
=−
p
∇·
U
+
ρJ
(2.39)
The terms in (2.37) that were eliminated by subtracting (2.38) represent the balance
of mechanical energy due to the motion of the fluid element; the remaining terms
represent the thermal energy balance.
Using the definition of geopotential (1.15), we have
g
Dz
D
Dt
gw
=
Dt
=
so that (2.38) can be rewritten as
1
2
U
D
Dt
ρ
·
U
+
=−
U
·∇
p
(2.40)
which is referred to as the
mechanical energy equation
. The sum of the kinetic
energy plus the gravitational potential energy is called the
mechanical energy
. Thus
(2.40) states that following the motion, the rate of change of mechanical energy
per unit volume equals the rate at which work is done by the pressure gradient
force.
The thermal energy equation (2.39) can be written in more familiar form by
noting from (2.31) that
1
ρ
∇·
1
ρ
2
Dρ
Dt
=
Dα
Dt
U
=−
and that for dry air the internal energy per unit mass is given by e
=
c
v
T , where
717Jkg
−
1
K
−
1
) is the specific heat at constant volume. We then obtain
c
v
(
=
DT
Dt
+
p
Dα
c
v
Dt
=
J
(2.41)
which is the usual form of the thermodynamic energy equation. Thus the first law
of thermodynamics indeed is applicable to a fluid in motion. The second term
on the left, representing the rate of working by the fluid system (per unit mass),
represents a conversion between thermal and mechanical energy. This conversion
process enables the solar heat energy to drive the motions of the atmosphere.
2.7
THERMODYNAMICS OF THE DRY ATMOSPHERE
Taking the total derivative of the equation of state (1.14), we obtain
p
Dα
α
Dp
R
DT
Dt
Dt
+
Dt
=
