Geography Reference
In-Depth Information
For synoptic scale motions the terms ∂u ∂x and ∂v ∂y tend to be of equal mag-
nitude but opposite sign. Thus, they tend to balance so that
∂u
∂x +
∂v
∂y
10 1 U
10 6 s 1
L
and in addition
∂w
∂z
W
H
10 6 s 1
Thus, terms B and C are each an order of magnitude greater than term A, and to a
first approximation, terms B and C balance in the continuity equation. To a good
approximation then
∂u
∂x +
∂v
∂y +
∂w
∂z +
d
dz (ln ρ 0 )
w
=
0
or, alternatively, in vector form
∇·
0 U )
=
0
(2.34)
Thus for synoptic scale motions the mass flux computed using the basic state den-
sity ρ 0 is nondivergent. This approximation is similar to the idealization of incom-
pressibility, which is often used in fluid mechanics. However, an incompressible
fluid has density constant following the motion:
Dt =
0
Thus by (2.31) the velocity divergence vanishes (
0) in an incompressible
fluid, which is not the same as (2.34). Our approximation (2.34) shows that for
purely horizontal flow the atmosphere behaves as though it were an incompressible
fluid. However, when there is vertical motion the compressibility associated with
the height dependence of ρ 0 must be taken into account.
∇·
U
=
2.6
THE THERMODYNAMIC ENERGY EQUATION
We now turn to the third fundamental conservation principle, the conservation of
energy as applied to a moving fluid element. The first law of thermodynamics is
usually derived by considering a system in thermodynamic equilibrium, that is,
a system that is initially at rest and after exchanging heat with its surroundings
and doing work on the surroundings is again at rest. For such a system the first
law states that the change in internal energy of the system is equal to the differ-
ence between the heat added to the system and the work done by the system .
 
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