Geography Reference
In-Depth Information
1.3
SCALE ANALYSIS
Scale analysis, or scaling, is a convenient technique for estimating the magnitudes
of various terms in the governing equations for a particular type of motion. In scal-
ing, typical expected values of the following quantities are specified:
(1) magnitudes of the field variables; (2) amplitudes of fluctuations in the field
variables; and (3) the characteristic length, depth, and time scales on which these
fluctuations occur. These typical values are then used to compare the magnitudes
of various terms in the governing equations. For example, in a typical midlatitude
synoptic
2
cyclone the surface pressure might fluctuate by 10 hPa over a horizontal
distance of 1000 km. Designating the amplitude of the horizontal pressure fluctu-
ation by δp, the horizontal coordinates by x and y, and the horizontal scale by L,
the magnitude of the horizontal pressure gradient may be estimated by dividing
δp by the length L to get
∂p
km
10
−
3
Pa m
−
1
∂x
,
∂p
δp
L
=
10 hpa/10
3
∼
∂y
Pressure fluctuations of similar magnitudes occur in other motion systems of vastly
different scale such as tornadoes, squall lines, and hurricanes. Thus, the horizon-
tal pressure gradient can range over several orders of magnitude for systems of
meteorological interest. Similar considerations are also valid for derivative terms
involving other field variables. Therefore, the nature of the dominant terms in the
governing equations is crucially dependent on the horizontal scale of the motions.
In particular, motions with horizontal scales of a few kilometers or less tend to
have short time scales so that terms involving the rotation of the earth are negli-
gible, while for larger scales they become very important. Because the character
of atmospheric motions depends so strongly on the horizontal scale, this scale
provides a convenient method for the classification of motion systems. Table 1.4
classifies examples of various types of motions by horizontal scale for the spectral
region from 10
−
7
to 10
7
m. In the following chapters, scaling arguments are used
extensively in developing simplifications of the governing equations for use in
modeling various types of motion systems.
1.4
FUNDAMENTAL FORCES
The motions of the atmosphere are governed by the fundamental physical laws
of conservation of mass, momentum, and energy. In Chapter 2, these principles
are applied to a small volume element of the atmosphere in order to obtain the
2
The term
synoptic
designates the branch of meteorology that deals with the analysis of observations
taken over a wide area at or near the same time. This term is commonly used (as here) to designate the
characteristic scale of the disturbances that are depicted on weather maps.
