Geography Reference
In-Depth Information
Equation (2.7) states that the acceleration following the motion in an inertial
system equals the rate of change of relative velocity following the relative motion
in the rotating frame plus the Coriolis acceleration due to relative motion in the
rotating frame plus the centripetal acceleration caused by the rotation of the coor-
dinates.
If we assume that the only real forces acting on the atmosphere are the pressure
gradient force, gravitation, and friction, we can rewrite Newton's second law (2.3)
with the aid of (2.7) as
D
U
Dt
=−
1
ρ
∇
p
+
2
×
U
−
g
+
F
r
(2.8)
where
F
r
designates the frictional force (see Section 1.4.3), and the centrifugal
force has been combined with gravitation in the gravity term
g
(see Section 1.5.2).
Equation (2.8) is the statement of Newton's second law for motion relative to
a rotating coordinate frame. It states that the acceleration following the relative
motion in the rotating frame equals the sum of the Coriolis force, the pressure
gradient force, effective gravity, and friction. This form of the momentum equation
is basic to most work in dynamic meteorology.
2.3
COMPONENT EQUATIONS IN SPHERICAL COORDINATES
For purposes of theoretical analysis and numerical prediction, it is necessary to
expand the vectorial momentum equation (2.8) into its scalar components. Since
the departure of the shape of the earth from sphericity is entirely negligible for
meteorological purposes, it is convenient to expand (2.8) in spherical coordinates
so that the (level) surface of the earth corresponds to a coordinate surface. The
coordinate axes are then (λ, φ, z), where λ is longitude, φ is latitude, and z is
the vertical distance above the surface of the earth. If the unit vectors
i
,
j
, and
k
are now taken to be directed eastward, northward, and upward, respectively, the
relative velocity becomes
≡
+
+
U
i
u
j
v
k
w
where the components u, v, and w are defined as
Dλ
Dt
,v
r
Dφ
Dz
Dt
u
≡
r cos φ
≡
Dt
,w
≡
(2.9)
Here, r is the distance to the center of the earth, which is related to z by r
z,
where a is the radius of the earth. Traditionally, the variable r in (2.9) is replaced by
the constant a. This is a very good approximation, as z
=
a
+
a for the regions of the
atmosphere with which meteorologists are concerned. For notational simplicity,
