Geography Reference
In-Depth Information
u
2
v
2
Dw
Dt
−
+
1
ρ
∂p
∂z
−
=−
g
+
2u cos φ
+
F
rz
(2.21)
a
which are the eastward, northward, and vertical component momentum equations,
respectively. The terms proportional to 1/a on the left-hand sides in (2.19)-(2.21)
are called
curvature
terms; they arise due to the curvature of the earth.
1
Because
they are nonlinear (i.e., they are quadratic in the dependent variables), they are dif-
ficult to handle in theoretical analyses. Fortunately, as shown in the next section,
the curvature terms are unimportant for midlatitude synoptic scale motions. How-
ever, even when the curvature terms are neglected, (2.19)-(2.21) are still nonlinear
partial differential equations, as can be seen by expanding the total derivatives into
their local and advective parts:
Du
Dt
=
∂u
∂t
+
u
∂u
v
∂u
w
∂u
∂z
∂x
+
∂y
+
with similar expressions for Dv/Dt and Dw/Dt . In general the advective accel-
eration terms are comparable in magnitude to the local acceleration. The presence
of nonlinear advection processes is one reason that dynamic meteorology is an
interesting and challenging subject.
2.4
SCALE ANALYSIS OF THE EQUATIONS OF MOTION
Section 1.3 discussed the basic notion of scaling the equations of motion in order
to determine whether some terms in the equations are negligible for motions of
meteorological concern. Elimination of terms on scaling considerations not only
has the advantage of simplifying the mathematics, but as shown in later chapters,
the elimination of small terms in some cases has the very important property of
completely eliminating or
filtering
an unwanted type of motion. The complete
equations of motion [(2.19)-(2.21)] describe all types and scales of atmospheric
motions. Sound waves, for example, are a perfectly valid class of solutions to
these equations. However, sound waves are of negligible importance in dynamical
meteorology. Therefore, it will be a distinct advantage if, as turns out to be true,
we can neglect the terms that lead to the production of sound waves and filter out
this unwanted class of motions.
In order to simplify (2.19)-(2.21) for synoptic scale motions, we define the
following characteristic scales of the field variables based on observed values for
midlatitude synoptic systems.
1
It can be shown that when r is replaced by a as done here (the traditional approximation) the
Coriolis terms proportional to cos φ in (2.19) and (2.21) must be neglected if the equations are to
satisfy angular momentum conservation.
