Geoscience Reference
In-Depth Information
Because the equations describing the movement and evolution of atmospheric
properties are complex, sometimes they can be written more efficiently using
shorthand representations, including using the 'summation convention' and
'vector algebra. Use of such representations has merit for the specialist because it
allows conciseness. However, their use requires first creating familiarity with
the efficient representation adopted, and to those who are not fluent in the
language of the selected representation this can inhibit ready understanding.
In this chapter, the primary goal is to convey understanding of how the basic
equations that describe the atmosphere arise. For this reason, use of such efficient
representations is largely avoided and clumsiness in representation is accepted
if this is more likely to improve comprehension. However, vector algebra equa-
tions are occasionally introduced when this is unlikely to confuse.
For reasons of conciseness, not all the equations sought are derived indepen-
dently and completely in this chapter. Often it is possible to derive an equation in
one dimension and its form in the other two dimensions follows by analogy. Also,
the need for some terms in one conservation equation can be easily recognized by
analogy with equivalent terms in another conservation equation.
Time rate of change in a fluid
Most readers will understand the distinction between the total derivative and
the partial derivative of a variable. However, for those less confident in this
branch of mathematics it is useful to review why the rate of change of an
atmospheric variable with time is written as the sum of more than one term. To
do this the rate of change of momentum in the direction of the mean wind
parallel to the ground is used as an example.
In this chapter the selected frame of reference is defined such that the Z axis
is perpendicular to and positive away from the ground, the X axis is parallel to
the ground pointing east along the line of latitude, and the Y axis is parallel to the
ground along the line of longitude. Consider the rate of change with time of
u
(the velocity along the X axis) at a point where the velocity of the moving air
has three components,
u
,
v
, and
w
along the X, Y, and Z axes respectively. For
simplicity, first consider the case when the air is only moving along the X axis.
Figure 16.1 illustrates that, in this case, there are two reasons why there may be a
change in the value of
u
at a particular point.
First, there may be a change in the value
u
at the point due to some (as yet
undefined) local force acting at that point (Fig. 16.1). This will give rise to a
change in the velocity component along the X axis which is represented by the
partial derivative of
u
with respect to time. However, even if there were no local
force acting, the air is moving past the point and the velocity along the X axis
within the moving body of air may not be constant. Consequently, the velocity in
the moving air as it passes at one time may be different to that within the air as it
passes at a later time (Fig. 16.1). The rate of change in the value of
u
resulting solely
