Geoscience Reference
In-Depth Information
An important distinction has to be made between variables that have both a mag-
nitude and a direction (like momentum or equivalently velocity) and variables that
have only a magnitude. The irst are called
vectors
, whereas the latter are referred to
as
scalars
(e.g., temperature, humidity, pressure).
Also note that we often talk about momentum (which is a three-dimensional vec-
tor) but actually refer to
horizontal
momentum along the mean low direction. This
quantity is a scalar.
cc
/
cc
/
−1
−
Question 3.3:
For an adiabatic process the following holds:
p
p
T
p
= constant
.
v
v
Thus both temperature and pressure change.
a) Do pressure and temperature change in the same direction (i.e., if one increases, the
other increases as well) or in opposite directions in an adiabatic process (values for
c
p
and
c
v
can be found in
Appendix B
)?
b) Use the equation of state for a perfect gas (gas law:
p T
=
ρ
) to deduce how the
density changes as a function of temperature in an adiabatic process.
c) The same steps as in (a) and (b) can also be used for the partial pressure of water
vapour (
e
):
e
cc
/
−−
=
1
cc
/
=
ρ
vv
. Hence, deduce
the dependence of
ρ
v
on temperature for an adiabatic process.
d) Show with the results of (b) and (c) that the speciic humidity indeed does not change
during adiabatic cooling.
p
T
p
constant in combination with
e
T
v
v
3.3.3 Statistical Description of Turbulence
In this section we discuss a number of statistical tools needed in the description of
turbulence and turbulent transport.
Reynolds Decomposition
Because turbulent lows are not reproducible in detail, we can treat them only in a
statistical sense (“how do things behave on average?”). The irst step in this statis-
tical description is the Reynolds decomposition (Reynolds,
1895
), which states that
a quantity
X
(might be a wind speed, temperature, etc.) at a given moment and at a
given location can be decomposed as:
X
=+′
X
X
(3.2)
where
X
is the mean value of
X
and
X
′ is the deviation from that mean. For
X
, in prin-
ciple only the so-called ensemble mean can be used. The ensemble mean requires that
one repeats an experiment (or natural situation) an ininite number of times, under
exactly the same conditions. Then the ensemble mean (at a given location and time) is
the mean over all those repetitions (thus the mean is space and time dependent). This
is - especially for natural systems outside the laboratory - impossible. Therefore,
the ensemble mean is generally approximated by a temporal mean (if observations
are made at a ixed position) or a spatial mean (if observations are made at different
positions).
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