Geoscience Reference
In-Depth Information
The flux of any entity is the average value of the product of the volume density
relevant to that entity with the velocity of the air in the required direction. For
example, the flux of water vapor in the X direction over a specified period is the
time-average of:
(
mass of water vapor per unit volume
)
×
(
velocity of air in X direction
)
This water vapor flux,
(m s
−1
), or kg
water
m
−2
s
−1
. Similarly, the sensible heat flux in the Y direction is the
time-average of:
, therefore has units of (kg m
−3
)
×
(kg
water
kg
−1
)
×
()
ρ
a
qu
(
heat content per unit volume of air
)
×
(
velocity in Y direction
)
Consequently, this sensible heat flux,
(
rq
ap
c v
)
, has units of (kg m
−3
)
×
(J kg
−1
K
−1
)
×
(K)
(m s
−1
), or J m
−2
s
−1
. And the flux of the Y component of momentum (recall
momentum is a vector) transferred in the Z direction is the time-average of:
×
(
momentum flux in the Y direction per unit volume of air
)
×
(
velocity in Z direction
)
Hence, this momentum flux,
()
r
a
uw
, has units of (kg m
−3
)
×
(m s
−1
)
×
(m s
−1
), or
kg m
−1
s
−2
.
However, such entities as '
heat content per unit volume of air
' and '
momentum in
the Y direction per unit volume of air
' are either rarely measured or they are
unmeasureable. Rather, it is the equivalent entities, namely '
virtual potential tem-
perature
' or '
velocity in the Y direction
' that are measured instead. A product of a
measurable atmospheric entity with a velocity component can thus be related to a
true flux with appropriate physical dimensions, but it has the advantage that it
appears naturally when cross products between atmospheric variables are defined.
Partly because of this, but also because (as shown later) doing so simplifies the
suite of equations that describe atmospheric flow and makes them more comparable
with each other, it is convenient to redefine the fluxes of mass, sensible heat,
momentum and moisture and minority constituents. This is done by dividing by the
density of moist air or, in the particular case of sensible heat, by the product of the
density of moist air with the specific heat of air. It is viable to do this because changes
in the density of air are typically 10% or less through the depth of the atmospheric
boundary layer. The equivalent flux so defined is called the
kinematic flux
.
Thus, each true flux expressed in appropriate physical units can be associated
with a kinematic flux in different units that is the cross product of a measurable
atmospheric variable with a measureable velocity component. Table 15.3 lists the
true fluxes and their units, the measurable atmospheric entities equivalent to each
flux, the relationship between each pair of actual and kinematic fluxes, and the
dimensions of the equivalent kinematic flux. The description of the theory of tur-
bulence that follows in later chapters is given using kinematic fluxes. However, it
is important to remember that each kinematic flux must ultimately be recast back
into a true flux in appropriate units, as for example when used in equations
describing surface energy exchange.
